MATH 114 - QUANTITATIVE
REASONING - Ratios and Proportions
Question Bank - Set 5
Liberty University
Question 1
Question
Simplify the following ratio: 2x2y3
3xy .
Solution
Step 1: To simplify the ratio, we can cancel out any common factors in the
numerator and denominator. Step 2: We can simplify the ratio by dividing both
the numerator and denominator by the highest common factor of the terms in
the ratio. Step 3: The highest common factor that can be cancelled out from
2x2y3and 3xy is xy. Step 4: Dividing 2x2y3by xy gives us 2x2y3
xy = 2xy2. Step
5: Dividing 3xy by xy gives us 3xy
xy = 3. Step 6: Therefore, the simplified ratio
2x2y3
3xy is equivalent to 2x2y3
3xy =2xy2
3.
So, 2x2y3
3xy simplifies to 2xy2
3.
Question 2
Question
A recipe calls for 4 cups of flour to make 24 cookies. If you want to make 36
cookies, how many cups of flour do you need?
Solution
Step 1: Determine the ratio of cups of flour to cookies in the original recipe.
Let xrepresent the number of cups of flour needed to make 36 cookies. We can
set up a proportion based on the given information:
4 cups of flour
24 cookies =xcups of flour
36 cookies
Step 2: Solve for xby cross-multiplying.
4×36 = 24x
144 = 24x
Step 3: Solve for xto find the number of cups of flour needed.
x=144
24 = 6
Therefore, you will need 6 cups of flour to make 36 cookies.
Question 3
Question
If 3x+ 2y= 12, find the ratio of xto ywhen x= 3y.
Solution
Let’s first substitute x= 3yinto the equation 3x+ 2y= 12 and solve for y.
Step 1: Substitute x= 3yinto the equation.
3(3y)+2y= 12
9y+ 2y= 12
11y= 12
y=12
11
Step 2: Find the value of xusing x= 3y.
x= 3(12
11) = 36
11
Step 3: Calculate the ratio of xto y.
x
y=
36
11
12
11
=36
12 = 3
So, the ratio of xto ywhen x= 3yis 3 .
2
Question 4
Question
A recipe for a fruit punch requires mixing 3 cups of orange juice, 4 cups of
pineapple juice, and 5 cups of cranberry juice. If a caterer wants to make a
smaller batch using 1 cup of orange juice, how many cups of pineapple juice and
cranberry juice should be used to maintain the same ratio of ingredients?
Solution
Let xrepresent the number of cups of pineapple juice needed, and yrepresent
the number of cups of cranberry juice needed to maintain the same ratio of
ingredients in the smaller batch.
Step 1: Set up the proportion using the ratios of the original recipe and
the smaller batch: 3
1=4
x=5
y
Step 2: Solve the first proportion for x:
3x= 4 ⇒x=4
3
Step 3: Solve the second proportion for y:
5y= 3 ⇒y=3
5
Step 4: Now that we have the ratios for the smaller batch, determine the
number of cups needed for pineapple juice and cranberry juice:
Pineapple juice = 4
3cups
Cranberry juice = 3
5cups
Therefore, to make a smaller batch with 1 cup of orange juice, the caterer
would need 4
3cups of pineapple juice and 3
5cups of cranberry juice.
Question 5
Question
In a certain class, the ratio of the number of male students to the number of
female students is 3:5. If there are 120 students in total, how many female
students are there in the class?
3
Solution
Let xrepresent the number of male students and yrepresent the number of
female students.
Step 1: Write the given information as a ratio. The ratio of male students
to female students is 3:5, so we have x:y= 3 : 5.
Step 2: Write an equation based on the total number of students. Since
there are 120 students in total, we have x+y= 120.
Step 3: Solve the system of equations. From the ratio x:y= 3 : 5, we can
write x=3
5y.
Substitute this expression for xinto the equation x+y= 120:
3
5y+y= 120
3
5y+5
5y= 120
8
5y= 120
8y= 120 ×5
8y= 600
y=600
8
y= 75
Step 4: Find the number of female students. There are 75 female students
in the class.
Question 6
Question
Solve the following proportion for x:
1
2x+ 3 =3x
4
4
Solution
To solve the given proportion, we can cross multiply to eliminate the fractions.
Step 1: Cross multiply to obtain:
4·1 = (2x+ 3) ·3x
Step 2: Simplify both sides of the equation:
4=6x2+ 9x
Step 3: Rearrange the equation to set it equal to zero:
6x2+ 9x−4=0
Step 4: To solve the quadratic equation, we can use the quadratic formula:
x=−b±√b2−4ac
2a
where a= 6, b= 9, and c=−4.
Step 5: Substitute the values of a,b, and cinto the quadratic formula and
solve for x:
x=−9±p92−4(6)(−4)
2(6)
x=−9±√81 + 96
12
x=−9±√177
12
Step 6: Therefore, the solutions for xare:
x=−9 + √177
12 or x=−9−√177
12
Question 7
Question
Solve for x:2
x+3 =5
2x−1.
5
Solution
Step 1: Cross-multiply to get rid of the fractions.
2
x+ 3 =5
2x−1
2(2x−1) = 5(x+ 3)
Step 2: Expand both sides of the equation.
4x−2=5x+ 15
Step 3: Rearrange the equation to isolate xon one side.
4x−5x= 15 + 2
−x= 17
Step 4: Solve for xby multiplying both sides by −1.
x=−17
Therefore, the solution to the equation is x=−17.
Question 8
Question
If the ratio of xto yis 3:5 and the ratio of yto zis 4:7, find the ratio of xto z.
Solution
Given: Ratio of xto y:x:y= 3 : 5
Ratio of yto z:y:z= 4 : 7
To find the ratio of xto z, we need to find a relation between xand zusing
the given information about x,y, and z.
Step 1: From the ratio of xto y, we can express yin terms of xas follows:
y=5
3x
Step 2: From the ratio of yto z, we can express yin terms of zas follows:
y=4
7z
Step 3: Equating the two expressions for y, we have:
5
3x=4
7z
6
Step 4: To find the ratio of xto z, solve for zin terms of x:
z=7
4×5
3x
Simplifying, we get:
z=35
12x
Step 5: Therefore, the ratio of xto zis x:z= 1 : 35
12 . Simplifying, we get
x:z= 12 : 35.
Thus, the ratio of xto zis 12:35.
Question 9
Question
If a mixture of paint is made by mixing red paint and blue paint in the ratio
3:5, how many liters of red paint should be mixed with 20 liters of blue paint
to make a mixture with a 40 liter ratio of red paint to blue paint?
Solution
Step 1: Find the amount of blue paint needed in the final mixture. Let xbe
the amount of red paint needed. Step 2: Write the ratio of red paint to blue
paint in the final mixture. Step 3: Set up the proportion using the ratios of red
paint to blue paint in the initial mixture and final mixture. Step 4: Solve the
proportion to find the amount of red paint needed.
Step 1:
Let xbe the amount of red paint needed in the final mixture.
Step 2:
The ratio of red paint to blue paint in the final mixture is 40:20 = 2:1.
Step 3:
Using the given ratios, we set up the proportion as follows:
3
5=x
20
2
1=x
20
7
Step 4:
Solving the proportion, we find:
3
5=x
20
3×20 = 5x
60 = 5x
x=60
5
x= 12
Therefore, 12 liters of red paint should be mixed with 20 liters of blue paint
to make a mixture with a 40 liter ratio of red paint to blue paint.
Question 10
Question
If 3 liters of water is mixed with 5 liters of alcohol, what is the ratio of water
to alcohol in the resulting mixture?
Solution
Step 1: Calculate the total volume of the resulting mixture.
Total volume = 3 liters + 5 liters = 8 liters
Step 2: Find the ratio of water to alcohol in the mixture. The ratio of water
to alcohol can be expressed as x:y. In this case, we will use 3xto represent the
volume of water and 5xto represent the volume of alcohol in the mixture.
3x+ 5x= 8
8x= 8
x= 1
Step 3: Therefore, the ratio of water to alcohol in the resulting mixture is:
3:5
8
Question 11
Question
Solve the proportion: 2x−5
3x+ 7 =8
11
Solution
Step 1: Cross multiply to eliminate the fractions:
(2x−5) ·11 = 8 ·(3x+ 7)
22x−55 = 24x+ 56
Step 2: Simplify the equation by collecting like terms:
22x−24x= 56 + 55
−2x= 111
Step 3: Divide by -2 to solve for x:
x=111
−2=−111
2
Therefore, the solution to the proportion is x=−111
2.
Question 12
Question
Solve the following proportion for x:
2x−3
x+ 5 =x+ 7
3x−2
Solution
Step 1: Cross multiply to get rid of the fractions.
(2x−3)(3x−2) = (x+ 5)(x+ 7)
6x2−4x−9x+ 6 = x2+ 7x+ 5x+ 35
6x2−13x+ 6 = x2+ 12x+ 35
Step 2: Simplify the equation and set it equal to zero.
6x2−13x+ 6 = x2+ 12x+ 35
5x2−25x−29 = 0
9
Step 3: Factor the quadratic equation.
5x2−25x−29 = 0
5(x2−5x)−29 = 0
5(x2−5x+25
4)−29 −5(25
4) = 0
5(x−5
2)2−29 −125
4= 0
5(x−5
2)2−181
4= 0
5(x−5
2)2=181
4
(x−5
2)2=181
20
Step 4: Solve for x.
x−5
2=±r181
20
x=5
2±r181
20
x=5±√181
2√5
x=5±√181
2√5·√5
√5
x=5√5±√905
10
Thus, the solutions for xare x=5√5+√905
10 and x=5√5−√905
10 .
Question 13
Question
If 3 men or 4 women can do a piece of work in 43 days. How long will 7 men
and 5 women together take to do the same work?
Solution
Step 1: Let’s assume that the work is represented by 1 unit. Thus, the work
rate of a man is 1
3×43 and the work rate of a woman is 1
4×43 .
Step 2: Next, we calculate the total work rate when 7 men and 5 women
work together:
71
3×43+ 5 1
4×43
10
Step 3: Simplifying the above expression, we get:
7
129 +5
172 =68
516 +45
516 =113
516
Step 4: Finally, we find the number of days required for 7 men and 5 women
to complete the work together by taking the reciprocal of the total work rate:
516
113 = 4 64
113 days
Therefore, 7 men and 5 women together will take 4 64
113 days to complete the
work.
Question 14
Question
A survey of 300 students showed that 45 of them like to play basketball. If
we assume that this sample is representative of the entire student population,
how many students in a school of 1500 students can we expect to like playing
basketball?
Solution
Step 1: Find the proportion of students who like to play basketball in the
sample. Step 2: Use the proportion to estimate the total number of students
who like playing basketball in the school of 1500 students.
Step 1: Calculate the proportion of students who like to play basketball in
the sample.
Proportion = Number of students who like basketball
Total number of students in the sample
Proportion = 45
300 =3
20
Step 2: Estimate the total number of students who like playing basketball
in the school of 1500 students.
Estimated number of students who like basketball = Proportion×Total number of students in the school
Estimated number of students who like basketball = 3
20 ×1500 = 225
Therefore, we can expect approximately 225 students in a school of 1500
students to like playing basketball.
11
Question 15
Question
If x,y, and zare three positive numbers such that x
y=3
2and y
z=5
4, find the
value of x+z
y.
Solution
Step 1: From the given ratios, we can set up the following equations:
x
y=3
2
y
z=5
4
Step 2: Rearranging the first equation, we have x=3
2y.
Step 3: Rearranging the second equation, we have z=4
5y.
Step 4: Substituting the expressions for xand zinto x+z
y, we get:
x+z
y=3
2y+4
5y
y
Step 5: Simplifying the expression inside the brackets, we have:
3
2y+4
5y
y=15
10 y+8
10 y
y=
23
10 y
y=23
10
Step 6: Therefore, the value of x+z
yis 23
10 .
Question 16
Question
Solve the proportion: 2x+8
3=x
6.
Solution
Step 1: Cross multiply to eliminate the fractions:
(2x+ 8) ×6=3×x
Step 2: Simplify both sides of the equation:
12x+ 48 = 3x
Step 3: Move all terms containing xto one side of the equation:
12x−3x=−48
12
Step 4: Simplify the left side of the equation:
9x=−48
Step 5: Solve for xby dividing both sides by 9:
x=−48
9=−16
3
Therefore, the solution to the proportion is x=−16
3.
Question 17
Question
If 12 workers can complete a project in 18 days, and 4 workers can complete
the same project in 54 days, how many more days would it take 8 workers to
complete the project compared to 12 workers?
Solution
Step 1: First, we find the work rate of each worker for both scenarios. Let r
be the work rate of each worker for the first scenario (12 workers in 18 days),
and r2be the work rate of each worker for the second scenario (4 workers in 54
days).
Step 2: For the first scenario, we have 12r×18 = 1 project, and for the
second scenario, we have 4r2×54 = 1 project. So, 12r=1
18 and 4r2=1
54 .
Step 3: Solve for rand r2.r=1
12×18 =1
216 and r2=1
4×54 =1
216 .
Step 4: Now, we find how long it takes 8 workers to complete the project.
Let dbe the number of days it takes 8 workers to complete the project.
Step 5: For 8 workers, we have 8 ×1
216 ×d= 1 project. Solving for d, we
get d=216
8.
Step 6: Calculate the number of days it takes 8 workers to complete the
project. d= 27 days.
Step 7: Finally, calculate the difference in days between 12 workers and 8
workers to complete the project. The difference is 27 −18 = 9 days. Therefore,
it would take 9 more days for 8 workers to complete the project compared to
12 workers.
Question 18
Question
If four quantities are in proportion, then adding 3 to the first and subtracting
2 from the second we get a new proportion. The terms in the new proportion
are 8, 9, 6, and 12. Find the original four quantities.
13
Solution
Let the original four quantities be a,b,c, and d, such that a
b=c
d. We are given
that after the transformation, the new proportion becomes a+3
b−2=c+3
d−2=8
6=
9
12 .
Step 1: Solve for aand busing the given new proportion.
Since a+3
b−2=8
6, we have:
a+ 3
b−2=8
6
a+ 3
b−2=4
3
3(a+ 3) = 4(b−2)
3a+ 9 = 4b−8
3a−4b=−17
Step 2: Solve for cand dusing the given new proportion.
Since c+3
d−2=9
12 , we have:
c+ 3
d−2=9
12
c+ 3
d−2=3
4
4(c+ 3) = 3(d−2)
4c+ 12 = 3d−6
4c−3d=−18
Step 3: Solve the system of equations to find the original quantities.
We need to solve the system of equations:
(3a−4b=−17
4c−3d=−18
Solving this system will give us the values of a,b,c, and d, which represent
the original four quantities.
Question 19
Question
Solve the following proportion for x:
3x
2=5
4−x
2
14
Solution
We will first simplify the proportion given and then solve for x.
Step 1: Simplify the proportion
3x
2=5
4−x
2
3x
2=5·2
4·2−x
2
3x
2=10
8−x
2
3x
2=5
4−x
2
3x
2=5−2x
4
Step 2: Cross multiply to solve for x
4·3x= 2 ·(5 −2x)
12x= 10 −4x
Step 3: Rearrange the equation to solve for x
12x+ 4x= 10
16x= 10
x=10
16
x=5
8
Therefore, the solution to the proportion is x=5
8.
Question 20
Question
Solve the proportion 2x+ 5
3x−4=7
8.
Solution
Step 1: Cross-multiply to obtain (2x+ 5) ·8 = 7 ·(3x−4). Step 2: Simplify
both sides to get 16x+40 = 21x−28. Step 3: Rearrange the equation to isolate
the variable by subtracting 16xfrom both sides and adding 28 to both sides:
40 + 28 = 21x−16x. Step 4: Simplify both sides to get 68 = 5x. Step 5:
Divide both sides by 5 to solve for x:x=68
5. Step 6: Thus, the solution to the
proportion is x=68
5.
15
Question 21
Question
If a car travels 180 miles in 3 hours, and another car travels 225 miles in 4 hours,
what is the ratio of their speeds?
Solution
Let’s denote the speed of the first car as s1and the speed of the second car as
s2. We can set up the following proportions based on the distances and times
given: 180 miles
3 hours =s1miles
1 hour and 225 miles
4 hours =s2miles
1 hour
From the first proportion, we find that s1= 60 mph, and from the second
proportion, we find that s2= 56.25 mph.
To find the ratio of their speeds, we divide the speed of the first car by the
speed of the second car:
s1
s2
=60 mph
56.25 mph =60
56.25 =80
75 =16
15
Therefore, the ratio of their speeds is 16 : 15 .
Question 22
Question
Solve for x:
4x−7
5x+ 9 =3x+ 2
2x−3
Solution
Step 1: Cross multiply to eliminate the fractions.
4x−7
5x+ 9 =3x+ 2
2x−3
⇒(4x−7)(2x−3) = (3x+ 2)(5x+ 9)
Step 2: Expand both sides of the equation.
8x2−12x−14x+ 21 = 15x2+ 27x+ 10x+ 18
⇒8x2−26x+ 21 = 15x2+ 37x+ 18
16
Step 3: Move terms to one side of the equation.
15x2+ 37x+ 18 −8x2+ 26x−21 = 0
7x2+ 63x−3=0
Step 4: Factor the quadratic equation.
(7x−1)(x+ 3) = 0
Step 5: Set each factor to zero to find the possible values of x.
7x−1 = 0 or x+ 3 = 0
⇒x=1
7or x=−3
Thus, the solutions are x=1
7or x=−3.
Question 23
Question
If 3 times the first of three consecutive odd integers is added to 4 times the
second, the result is 11 more than 5 times the third. Find the integers.
Solution
Let’s denote the three consecutive odd integers as x,x+ 2, and x+ 4.
Step 1: Set up the equation based on the given information. We
are given that 3x+ 4(x+ 2) = 5(x+ 4) + 11. We can simplify this equation to
solve for x.
Step 2: Solve for x.Expanding and simplifying the equation:
3x+ 4x+ 8 = 5x+ 20 + 11
7x+ 8 = 5x+ 31
7x−5x= 31 −8
2x= 23
x=23
2
x= 11.5
Since xshould be an integer, we made a mistake. Hence, there is no solution to
this problem.
17
Question 24
Question
A recipe for a cake calls for 3 cups of flour for every 1 cup of sugar. If you want
to make a cake using 9 cups of flour, how many cups of sugar should you use?
Solution
Step 1: Let’s set up a proportion using the information given in the question.
Let xbe the number of cups of sugar needed. We can set up the proportion:
3
1=9
x.
Step 2: Solve the proportion for x. Cross multiplying, we get 3x= 9. Divide
both sides by 3 to solve for x:x=9
3= 3.
Step 3: Therefore, you should use 3 cups of sugar when making the cake to
go with 9 cups of flour in the recipe.
Question 25
Question
A recipe for muffins calls for 2 cups of flour and 1 cup of sugar. If you want to
make 18 muffins instead of the original 12, how much flour should you use?
Solution
Step 1: First, let’s find the ratio of flour to sugar in the original recipe. - The
ratio of flour to sugar in the original recipe is 2 : 1.
Step 2: Next, we need to find the ratio of flour to sugar required to make 18
muffins. - Since we are increasing the number of muffins from 12 to 18, the new
ratio of flour to sugar would be x: 1, where xis the amount of flour needed for
18 muffins.
Step 3: Set up a proportion using the ratios from the original recipe and the
desired recipe for 18 muffins. 2
1=x
1
Step 4: Solve for x.
2 = x⇒x= 2
Step 5: Therefore, to make 18 muffins, you will need 2 cups of flour.
Question 26
Question
Solve the following proportion for x:
18
2x+ 3
4x−1=5
7
Solution
Step 1: Cross multiply to eliminate the fractions.
(2x+ 3) ·7=5·(4x−1)
Step 2: Expand both sides of the equation.
14x+ 21 = 20x−5
Step 3: Rearrange the equation by moving all terms involving xto one side.
14x+ 21 = 20x−5
14x−20x=−5−21
Step 4: Combine like terms.
−6x=−26
Step 5: Solve for xby dividing both sides by −6.
x=−26
−6
Step 6: Simplify the fraction to find the final answer.
x=13
3
Question 27
Question
If 2x+ 3y= 12 and x
2=y
3, find the values of xand y.
Solution
Step 1: Solve the second equation for xin terms of y.
x
2=y
3=⇒x=2y
3
Step 2: Substitute the expression for xinto the first equation and solve for
y.
22y
3+ 3y= 12
19
4y
3+ 3y= 12
4y+ 9y
3= 12
13y
3= 12
13y= 36 =⇒y=36
13
Step 3: Substitute the value of yback into the expression for xto find its
value.
x=2(36
13 )
3=72
39 =24
13
Therefore, the values of xand yare x=24
13 and y=36
13 .
Question 28
Question
If x:y= 4 : 3 and y:z= 5 : 6, find x:y:z.
Solution
Step 1: Let’s find a common ratio that relates x,y, and zby considering the
given ratios.
Step 2: Since x:y= 4 : 3 and y:z= 5 : 6, we can rewrite these ratios using
a common term. We do this by setting up a proportionality equation:
x
y=4
3and y
z=5
6
Step 3: By cross-multiplying in both equations, we get:
3x= 4yand 5y= 6z
Step 4: Solving for yin terms of xand z:
y=3
4xand y=6
5z
Step 5: Equating these two expressions for y, we have:
3
4x=6
5z
Step 6: Simplifying this equation to solve for zin terms of x:
z=5
4x
Step 7: Now we have the ratios x:y= 4 : 3 and y:z= 5 : 6, which can
be expressed as x:y:z= 20 : 15 : 18. Thus, the ratios of x,y, and zare
20 : 15 : 18.
20
Question 29
Question
Solve for x:3x−10
2x+5 =x+2
3x−6.
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x−10)(3x−6) = (2x+ 5)(x+ 2)
9x2−18x−30x+ 60 = 2x2+ 5x+ 4x+ 10
9x2−48x+ 60 = 2x2+ 9x+ 10
Step 2: Combine like terms and move all terms to one side of the equation.
9x2−48x+ 60 −2x2−9x−10 = 0
7x2−57x+ 50 = 0
Step 3: Factor the quadratic equation.
(7x−5)(x−10) = 0
Step 4: Set each factor to zero and solve for x.
7x−5 = 0 (or) x−10 = 0
7x= 5 (or) x= 10
x=5
7(or) x= 10
Therefore, the solutions are x=5
7or x= 10.
Question 30
Question
Solve the following proportion for x:
2
3=4
x.
Solution
To solve the proportion 2
3=4
xfor x, we can cross multiply.
Step 1: Cross multiply to get:
2·x= 3 ·4.
21
Step 2: Simplify both sides of the equation:
2x= 12.
Step 3: To solve for x, divide both sides by 2:
2x
2=12
2.
Step 4: Simplify to find the value of x:
x= 6.
Therefore, the solution to the proportion 2
3=4
xis x= 6.
Question 31
Question
Simplify the following expression involving ratios and proportions:
3
5∇ · 1 + 2
3
Solution
Step 1: We start by simplifying the expression within the parentheses. Step 2:
We add 1 and 2
3. Step 3: This gives us 1 + 2
3=3
3+2
3=5
3. Step 4: Next, we
rewrite the expression with the simplified parentheses:
3
5∇ · 5
3
Step 5: To divide fractions, we multiply by the reciprocal of the denominator.
Step 6: This gives us 3
5×3
5. Step 7: We multiply the numerators together
and the denominators together. Step 8: So, 3×3
5×5=9
25 . Step 9: Therefore, the
simplified expression is 9
25 .
Question 32
Question
If 4 workers can finish a job in 6 days, how many workers are needed to finish
the same job in 3 days?
22
Solution
Step 1: Let’s first calculate the total work required to finish the job. Given that
4 workers can finish the job in 6 days, we can find the amount of work done by
each worker in 1 day. Let xbe the amount of work done by each worker in 1
day. So the total work required to finish the job is 4 ×xin 6 days.
Total work = 4x×6 = 24x
Step 2: Now, we need to find out how many workers are needed to finish the
job in 3 days. Let ybe the number of workers required to finish the job in 3
days. Since the total work remains the same, we can set up a proportion:
4x×6 = y×3
24x= 3y
Step 3: We know that x
y=1
3. So, to find the number of workers required,
we need to solve the equation x
y=1
3for y.
x
y=1
3
3x=y
Therefore, to finish the same job in 3 days, we would need 3 workers.
Question 33
Question
If 3 tablespoons of sugar are needed to make 12 cookies, how many tablespoons
of sugar are needed to make 72 cookies?
Solution
Step 1: Let xrepresent the number of tablespoons of sugar needed to make 72
cookies. We can set up a proportion using the given information:
3
12 =x
72
Step 2: Solve for xby cross multiplying:
3×72 = 12 ×x
216 = 12x
Step 3: Divide both sides by 12 to solve for x:
x=216
12 = 18
Therefore, 18 tablespoons of sugar are needed to make 72 cookies.
23
Question 34
Question
Solve the following proportion for x:
2
x+ 3 =x+ 1
4
Solution
Step 1: Cross multiply to eliminate the fractions.
2·4=(x+ 3) ·(x+ 1)
8 = x2+ 4x+ 3
Step 2: Rearrange the equation into standard form.
x2+ 4x+ 3 −8=0
x2+ 4x−5 = 0
Step 3: Factor the quadratic equation.
x2+ 4x−5=(x+ 5)(x−1) = 0
Step 4: Set each factor to zero and solve for x.
x+ 5 = 0 (Case 1)
x=−5
x−1 = 0 (Case 2)
x= 1
Final Answer: x=−5 or x= 1.
Question 35
Question
A bag contains some red balls and blue balls. The ratio of red balls to blue balls
is 2:3. If the total number of balls in the bag is 100, how many of them are red?
Solution
Step 1: Let’s set up equations based on the given information. Let xrepresent
the number of red balls and yrepresent the number of blue balls. We have the
following:
ratio of red balls to blue balls = 2 : 3
total number of balls = 100
24
Step 2: From the ratio, we can write the following equation:
x
y=2
3
Step 3: We can also write an equation based on the total number of balls:
x+y= 100
Step 4: To solve these two equations simultaneously, we can first rewrite the
ratio equation as:
3x= 2y
Step 5: Now, we can use substitution or elimination to find the values of x
and y. Let’s use elimination. Multiply the second equation by 3 and add it to
the third equation:
(3x= 2y
3(x+y) = 300
Step 6: Simplify the equations:
(3x= 2y
3x+ 3y= 300
Step 7: Subtract the first equation from the second equation:
3y−2y= 300
y= 100
Step 8: Now that we have found the number of blue balls, we can find the
number of red balls by substituting back into the third equation:
x+ 100 = 100
x= 0
Step 9: Therefore, there are 0 red balls in the bag.
25
Solution
Step 1: Determine the ratio of cups of flour to cookies in the original recipe.
Let xrepresent the number of cups of flour needed to make 36 cookies. We can
set up a proportion based on the given information:
4 cups of flour
24 cookies =xcups of flour
36 cookies
Step 2: Solve for xby cross-multiplying.
4×36 = 24x
144 = 24x
Step 3: Solve for xto find the number of cups of flour needed.
x=144
24 = 6
Therefore, you will need 6 cups of flour to make 36 cookies.
Question 3
Question
If 3x+ 2y= 12, find the ratio of xto ywhen x= 3y.
Solution
Let’s first substitute x= 3yinto the equation 3x+ 2y= 12 and solve for y.
Step 1: Substitute x= 3yinto the equation.
3(3y)+2y= 12
9y+ 2y= 12
11y= 12
y=12
11
Step 2: Find the value of xusing x= 3y.
x= 3(12
11) = 36
11
Step 3: Calculate the ratio of xto y.
x
y=
36
11
12
11
=36
12 = 3
So, the ratio of xto ywhen x= 3yis 3 .
2
Question 4
Question
A recipe for a fruit punch requires mixing 3 cups of orange juice, 4 cups of
pineapple juice, and 5 cups of cranberry juice. If a caterer wants to make a
smaller batch using 1 cup of orange juice, how many cups of pineapple juice and
cranberry juice should be used to maintain the same ratio of ingredients?
Solution
Let xrepresent the number of cups of pineapple juice needed, and yrepresent
the number of cups of cranberry juice needed to maintain the same ratio of
ingredients in the smaller batch.
Step 1: Set up the proportion using the ratios of the original recipe and
the smaller batch: 3
1=4
x=5
y
Step 2: Solve the first proportion for x:
3x= 4 ⇒x=4
3
Step 3: Solve the second proportion for y:
5y= 3 ⇒y=3
5
Step 4: Now that we have the ratios for the smaller batch, determine the
number of cups needed for pineapple juice and cranberry juice:
Pineapple juice = 4
3cups
Cranberry juice = 3
5cups
Therefore, to make a smaller batch with 1 cup of orange juice, the caterer
would need 4
3cups of pineapple juice and 3
5cups of cranberry juice.
Question 5
Question
In a certain class, the ratio of the number of male students to the number of
female students is 3:5. If there are 120 students in total, how many female
students are there in the class?
3
Solution
Let xrepresent the number of male students and yrepresent the number of
female students.
Step 1: Write the given information as a ratio. The ratio of male students
to female students is 3:5, so we have x:y= 3 : 5.
Step 2: Write an equation based on the total number of students. Since
there are 120 students in total, we have x+y= 120.
Step 3: Solve the system of equations. From the ratio x:y= 3 : 5, we can
write x=3
5y.
Substitute this expression for xinto the equation x+y= 120:
3
5y+y= 120
3
5y+5
5y= 120
8
5y= 120
8y= 120 ×5
8y= 600
y=600
8
y= 75
Step 4: Find the number of female students. There are 75 female students
in the class.
Question 6
Question
Solve the following proportion for x:
1
2x+ 3 =3x
4
4
Solution
To solve the given proportion, we can cross multiply to eliminate the fractions.
Step 1: Cross multiply to obtain:
4·1 = (2x+ 3) ·3x
Step 2: Simplify both sides of the equation:
4=6x2+ 9x
Step 3: Rearrange the equation to set it equal to zero:
6x2+ 9x−4=0
Step 4: To solve the quadratic equation, we can use the quadratic formula:
x=−b±√b2−4ac
2a
where a= 6, b= 9, and c=−4.
Step 5: Substitute the values of a,b, and cinto the quadratic formula and
solve for x:
x=−9±p92−4(6)(−4)
2(6)
x=−9±√81 + 96
12
x=−9±√177
12
Step 6: Therefore, the solutions for xare:
x=−9 + √177
12 or x=−9−√177
12
Question 7
Question
Solve for x:2
x+3 =5
2x−1.
5
Solution
Step 1: Cross-multiply to get rid of the fractions.
2
x+ 3 =5
2x−1
2(2x−1) = 5(x+ 3)
Step 2: Expand both sides of the equation.
4x−2=5x+ 15
Step 3: Rearrange the equation to isolate xon one side.
4x−5x= 15 + 2
−x= 17
Step 4: Solve for xby multiplying both sides by −1.
x=−17
Therefore, the solution to the equation is x=−17.
Question 8
Question
If the ratio of xto yis 3:5 and the ratio of yto zis 4:7, find the ratio of xto z.
Solution
Given: Ratio of xto y:x:y= 3 : 5
Ratio of yto z:y:z= 4 : 7
To find the ratio of xto z, we need to find a relation between xand zusing
the given information about x,y, and z.
Step 1: From the ratio of xto y, we can express yin terms of xas follows:
y=5
3x
Step 2: From the ratio of yto z, we can express yin terms of zas follows:
y=4
7z
Step 3: Equating the two expressions for y, we have:
5
3x=4
7z
6
Step 4: To find the ratio of xto z, solve for zin terms of x:
z=7
4×5
3x
Simplifying, we get:
z=35
12x
Step 5: Therefore, the ratio of xto zis x:z= 1 : 35
12 . Simplifying, we get
x:z= 12 : 35.
Thus, the ratio of xto zis 12:35.
Question 9
Question
If a mixture of paint is made by mixing red paint and blue paint in the ratio
3:5, how many liters of red paint should be mixed with 20 liters of blue paint
to make a mixture with a 40 liter ratio of red paint to blue paint?
Solution
Step 1: Find the amount of blue paint needed in the final mixture. Let xbe
the amount of red paint needed. Step 2: Write the ratio of red paint to blue
paint in the final mixture. Step 3: Set up the proportion using the ratios of red
paint to blue paint in the initial mixture and final mixture. Step 4: Solve the
proportion to find the amount of red paint needed.
Step 1:
Let xbe the amount of red paint needed in the final mixture.
Step 2:
The ratio of red paint to blue paint in the final mixture is 40:20 = 2:1.
Step 3:
Using the given ratios, we set up the proportion as follows:
3
5=x
20
2
1=x
20
7
Step 4:
Solving the proportion, we find:
3
5=x
20
3×20 = 5x
60 = 5x
x=60
5
x= 12
Therefore, 12 liters of red paint should be mixed with 20 liters of blue paint
to make a mixture with a 40 liter ratio of red paint to blue paint.
Question 10
Question
If 3 liters of water is mixed with 5 liters of alcohol, what is the ratio of water
to alcohol in the resulting mixture?
Solution
Step 1: Calculate the total volume of the resulting mixture.
Total volume = 3 liters + 5 liters = 8 liters
Step 2: Find the ratio of water to alcohol in the mixture. The ratio of water
to alcohol can be expressed as x:y. In this case, we will use 3xto represent the
volume of water and 5xto represent the volume of alcohol in the mixture.
3x+ 5x= 8
8x= 8
x= 1
Step 3: Therefore, the ratio of water to alcohol in the resulting mixture is:
3:5
8
Question 11
Question
Solve the proportion: 2x−5
3x+ 7 =8
11
Solution
Step 1: Cross multiply to eliminate the fractions:
(2x−5) ·11 = 8 ·(3x+ 7)
22x−55 = 24x+ 56
Step 2: Simplify the equation by collecting like terms:
22x−24x= 56 + 55
−2x= 111
Step 3: Divide by -2 to solve for x:
x=111
−2=−111
2
Therefore, the solution to the proportion is x=−111
2.
Question 12
Question
Solve the following proportion for x:
2x−3
x+ 5 =x+ 7
3x−2
Solution
Step 1: Cross multiply to get rid of the fractions.
(2x−3)(3x−2) = (x+ 5)(x+ 7)
6x2−4x−9x+ 6 = x2+ 7x+ 5x+ 35
6x2−13x+ 6 = x2+ 12x+ 35
Step 2: Simplify the equation and set it equal to zero.
6x2−13x+ 6 = x2+ 12x+ 35
5x2−25x−29 = 0
9
Step 3: Factor the quadratic equation.
5x2−25x−29 = 0
5(x2−5x)−29 = 0
5(x2−5x+25
4)−29 −5(25
4) = 0
5(x−5
2)2−29 −125
4= 0
5(x−5
2)2−181
4= 0
5(x−5
2)2=181
4
(x−5
2)2=181
20
Step 4: Solve for x.
x−5
2=±r181
20
x=5
2±r181
20
x=5±√181
2√5
x=5±√181
2√5·√5
√5
x=5√5±√905
10
Thus, the solutions for xare x=5√5+√905
10 and x=5√5−√905
10 .
Question 13
Question
If 3 men or 4 women can do a piece of work in 43 days. How long will 7 men
and 5 women together take to do the same work?
Solution
Step 1: Let’s assume that the work is represented by 1 unit. Thus, the work
rate of a man is 1
3×43 and the work rate of a woman is 1
4×43 .
Step 2: Next, we calculate the total work rate when 7 men and 5 women
work together:
71
3×43+ 5 1
4×43
10
Step 3: Simplifying the above expression, we get:
7
129 +5
172 =68
516 +45
516 =113
516
Step 4: Finally, we find the number of days required for 7 men and 5 women
to complete the work together by taking the reciprocal of the total work rate:
516
113 = 4 64
113 days
Therefore, 7 men and 5 women together will take 4 64
113 days to complete the
work.
Question 14
Question
A survey of 300 students showed that 45 of them like to play basketball. If
we assume that this sample is representative of the entire student population,
how many students in a school of 1500 students can we expect to like playing
basketball?
Solution
Step 1: Find the proportion of students who like to play basketball in the
sample. Step 2: Use the proportion to estimate the total number of students
who like playing basketball in the school of 1500 students.
Step 1: Calculate the proportion of students who like to play basketball in
the sample.
Proportion = Number of students who like basketball
Total number of students in the sample
Proportion = 45
300 =3
20
Step 2: Estimate the total number of students who like playing basketball
in the school of 1500 students.
Estimated number of students who like basketball = Proportion×Total number of students in the school
Estimated number of students who like basketball = 3
20 ×1500 = 225
Therefore, we can expect approximately 225 students in a school of 1500
students to like playing basketball.
11
Question 15
Question
If x,y, and zare three positive numbers such that x
y=3
2and y
z=5
4, find the
value of x+z
y.
Solution
Step 1: From the given ratios, we can set up the following equations:
x
y=3
2
y
z=5
4
Step 2: Rearranging the first equation, we have x=3
2y.
Step 3: Rearranging the second equation, we have z=4
5y.
Step 4: Substituting the expressions for xand zinto x+z
y, we get:
x+z
y=3
2y+4
5y
y
Step 5: Simplifying the expression inside the brackets, we have:
3
2y+4
5y
y=15
10 y+8
10 y
y=
23
10 y
y=23
10
Step 6: Therefore, the value of x+z
yis 23
10 .
Question 16
Question
Solve the proportion: 2x+8
3=x
6.
Solution
Step 1: Cross multiply to eliminate the fractions:
(2x+ 8) ×6=3×x
Step 2: Simplify both sides of the equation:
12x+ 48 = 3x
Step 3: Move all terms containing xto one side of the equation:
12x−3x=−48
12
Step 4: Simplify the left side of the equation:
9x=−48
Step 5: Solve for xby dividing both sides by 9:
x=−48
9=−16
3
Therefore, the solution to the proportion is x=−16
3.
Question 17
Question
If 12 workers can complete a project in 18 days, and 4 workers can complete
the same project in 54 days, how many more days would it take 8 workers to
complete the project compared to 12 workers?
Solution
Step 1: First, we find the work rate of each worker for both scenarios. Let r
be the work rate of each worker for the first scenario (12 workers in 18 days),
and r2be the work rate of each worker for the second scenario (4 workers in 54
days).
Step 2: For the first scenario, we have 12r×18 = 1 project, and for the
second scenario, we have 4r2×54 = 1 project. So, 12r=1
18 and 4r2=1
54 .
Step 3: Solve for rand r2.r=1
12×18 =1
216 and r2=1
4×54 =1
216 .
Step 4: Now, we find how long it takes 8 workers to complete the project.
Let dbe the number of days it takes 8 workers to complete the project.
Step 5: For 8 workers, we have 8 ×1
216 ×d= 1 project. Solving for d, we
get d=216
8.
Step 6: Calculate the number of days it takes 8 workers to complete the
project. d= 27 days.
Step 7: Finally, calculate the difference in days between 12 workers and 8
workers to complete the project. The difference is 27 −18 = 9 days. Therefore,
it would take 9 more days for 8 workers to complete the project compared to
12 workers.
Question 18
Question
If four quantities are in proportion, then adding 3 to the first and subtracting
2 from the second we get a new proportion. The terms in the new proportion
are 8, 9, 6, and 12. Find the original four quantities.
13
Solution
Let the original four quantities be a,b,c, and d, such that a
b=c
d. We are given
that after the transformation, the new proportion becomes a+3
b−2=c+3
d−2=8
6=
9
12 .
Step 1: Solve for aand busing the given new proportion.
Since a+3
b−2=8
6, we have:
a+ 3
b−2=8
6
a+ 3
b−2=4
3
3(a+ 3) = 4(b−2)
3a+ 9 = 4b−8
3a−4b=−17
Step 2: Solve for cand dusing the given new proportion.
Since c+3
d−2=9
12 , we have:
c+ 3
d−2=9
12
c+ 3
d−2=3
4
4(c+ 3) = 3(d−2)
4c+ 12 = 3d−6
4c−3d=−18
Step 3: Solve the system of equations to find the original quantities.
We need to solve the system of equations:
(3a−4b=−17
4c−3d=−18
Solving this system will give us the values of a,b,c, and d, which represent
the original four quantities.
Question 19
Question
Solve the following proportion for x:
3x
2=5
4−x
2
14
Solution
We will first simplify the proportion given and then solve for x.
Step 1: Simplify the proportion
3x
2=5
4−x
2
3x
2=5·2
4·2−x
2
3x
2=10
8−x
2
3x
2=5
4−x
2
3x
2=5−2x
4
Step 2: Cross multiply to solve for x
4·3x= 2 ·(5 −2x)
12x= 10 −4x
Step 3: Rearrange the equation to solve for x
12x+ 4x= 10
16x= 10
x=10
16
x=5
8
Therefore, the solution to the proportion is x=5
8.
Question 20
Question
Solve the proportion 2x+ 5
3x−4=7
8.
Solution
Step 1: Cross-multiply to obtain (2x+ 5) ·8 = 7 ·(3x−4). Step 2: Simplify
both sides to get 16x+40 = 21x−28. Step 3: Rearrange the equation to isolate
the variable by subtracting 16xfrom both sides and adding 28 to both sides:
40 + 28 = 21x−16x. Step 4: Simplify both sides to get 68 = 5x. Step 5:
Divide both sides by 5 to solve for x:x=68
5. Step 6: Thus, the solution to the
proportion is x=68
5.
15
Question 21
Question
If a car travels 180 miles in 3 hours, and another car travels 225 miles in 4 hours,
what is the ratio of their speeds?
Solution
Let’s denote the speed of the first car as s1and the speed of the second car as
s2. We can set up the following proportions based on the distances and times
given: 180 miles
3 hours =s1miles
1 hour and 225 miles
4 hours =s2miles
1 hour
From the first proportion, we find that s1= 60 mph, and from the second
proportion, we find that s2= 56.25 mph.
To find the ratio of their speeds, we divide the speed of the first car by the
speed of the second car:
s1
s2
=60 mph
56.25 mph =60
56.25 =80
75 =16
15
Therefore, the ratio of their speeds is 16 : 15 .
Question 22
Question
Solve for x:
4x−7
5x+ 9 =3x+ 2
2x−3
Solution
Step 1: Cross multiply to eliminate the fractions.
4x−7
5x+ 9 =3x+ 2
2x−3
⇒(4x−7)(2x−3) = (3x+ 2)(5x+ 9)
Step 2: Expand both sides of the equation.
8x2−12x−14x+ 21 = 15x2+ 27x+ 10x+ 18
⇒8x2−26x+ 21 = 15x2+ 37x+ 18
16
Step 3: Move terms to one side of the equation.
15x2+ 37x+ 18 −8x2+ 26x−21 = 0
7x2+ 63x−3=0
Step 4: Factor the quadratic equation.
(7x−1)(x+ 3) = 0
Step 5: Set each factor to zero to find the possible values of x.
7x−1 = 0 or x+ 3 = 0
⇒x=1
7or x=−3
Thus, the solutions are x=1
7or x=−3.
Question 23
Question
If 3 times the first of three consecutive odd integers is added to 4 times the
second, the result is 11 more than 5 times the third. Find the integers.
Solution
Let’s denote the three consecutive odd integers as x,x+ 2, and x+ 4.
Step 1: Set up the equation based on the given information. We
are given that 3x+ 4(x+ 2) = 5(x+ 4) + 11. We can simplify this equation to
solve for x.
Step 2: Solve for x.Expanding and simplifying the equation:
3x+ 4x+ 8 = 5x+ 20 + 11
7x+ 8 = 5x+ 31
7x−5x= 31 −8
2x= 23
x=23
2
x= 11.5
Since xshould be an integer, we made a mistake. Hence, there is no solution to
this problem.
17
Question 24
Question
A recipe for a cake calls for 3 cups of flour for every 1 cup of sugar. If you want
to make a cake using 9 cups of flour, how many cups of sugar should you use?
Solution
Step 1: Let’s set up a proportion using the information given in the question.
Let xbe the number of cups of sugar needed. We can set up the proportion:
3
1=9
x.
Step 2: Solve the proportion for x. Cross multiplying, we get 3x= 9. Divide
both sides by 3 to solve for x:x=9
3= 3.
Step 3: Therefore, you should use 3 cups of sugar when making the cake to
go with 9 cups of flour in the recipe.
Question 25
Question
A recipe for muffins calls for 2 cups of flour and 1 cup of sugar. If you want to
make 18 muffins instead of the original 12, how much flour should you use?
Solution
Step 1: First, let’s find the ratio of flour to sugar in the original recipe. - The
ratio of flour to sugar in the original recipe is 2 : 1.
Step 2: Next, we need to find the ratio of flour to sugar required to make 18
muffins. - Since we are increasing the number of muffins from 12 to 18, the new
ratio of flour to sugar would be x: 1, where xis the amount of flour needed for
18 muffins.
Step 3: Set up a proportion using the ratios from the original recipe and the
desired recipe for 18 muffins. 2
1=x
1
Step 4: Solve for x.
2 = x⇒x= 2
Step 5: Therefore, to make 18 muffins, you will need 2 cups of flour.
Question 26
Question
Solve the following proportion for x:
18
2x+ 3
4x−1=5
7
Solution
Step 1: Cross multiply to eliminate the fractions.
(2x+ 3) ·7=5·(4x−1)
Step 2: Expand both sides of the equation.
14x+ 21 = 20x−5
Step 3: Rearrange the equation by moving all terms involving xto one side.
14x+ 21 = 20x−5
14x−20x=−5−21
Step 4: Combine like terms.
−6x=−26
Step 5: Solve for xby dividing both sides by −6.
x=−26
−6
Step 6: Simplify the fraction to find the final answer.
x=13
3
Question 27
Question
If 2x+ 3y= 12 and x
2=y
3, find the values of xand y.
Solution
Step 1: Solve the second equation for xin terms of y.
x
2=y
3=⇒x=2y
3
Step 2: Substitute the expression for xinto the first equation and solve for
y.
22y
3+ 3y= 12
19
4y
3+ 3y= 12
4y+ 9y
3= 12
13y
3= 12
13y= 36 =⇒y=36
13
Step 3: Substitute the value of yback into the expression for xto find its
value.
x=2(36
13 )
3=72
39 =24
13
Therefore, the values of xand yare x=24
13 and y=36
13 .
Question 28
Question
If x:y= 4 : 3 and y:z= 5 : 6, find x:y:z.
Solution
Step 1: Let’s find a common ratio that relates x,y, and zby considering the
given ratios.
Step 2: Since x:y= 4 : 3 and y:z= 5 : 6, we can rewrite these ratios using
a common term. We do this by setting up a proportionality equation:
x
y=4
3and y
z=5
6
Step 3: By cross-multiplying in both equations, we get:
3x= 4yand 5y= 6z
Step 4: Solving for yin terms of xand z:
y=3
4xand y=6
5z
Step 5: Equating these two expressions for y, we have:
3
4x=6
5z
Step 6: Simplifying this equation to solve for zin terms of x:
z=5
4x
Step 7: Now we have the ratios x:y= 4 : 3 and y:z= 5 : 6, which can
be expressed as x:y:z= 20 : 15 : 18. Thus, the ratios of x,y, and zare
20 : 15 : 18.
20
Question 29
Question
Solve for x:3x−10
2x+5 =x+2
3x−6.
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x−10)(3x−6) = (2x+ 5)(x+ 2)
9x2−18x−30x+ 60 = 2x2+ 5x+ 4x+ 10
9x2−48x+ 60 = 2x2+ 9x+ 10
Step 2: Combine like terms and move all terms to one side of the equation.
9x2−48x+ 60 −2x2−9x−10 = 0
7x2−57x+ 50 = 0
Step 3: Factor the quadratic equation.
(7x−5)(x−10) = 0
Step 4: Set each factor to zero and solve for x.
7x−5 = 0 (or) x−10 = 0
7x= 5 (or) x= 10
x=5
7(or) x= 10
Therefore, the solutions are x=5
7or x= 10.
Question 30
Question
Solve the following proportion for x:
2
3=4
x.
Solution
To solve the proportion 2
3=4
xfor x, we can cross multiply.
Step 1: Cross multiply to get:
2·x= 3 ·4.
21
Step 2: Simplify both sides of the equation:
2x= 12.
Step 3: To solve for x, divide both sides by 2:
2x
2=12
2.
Step 4: Simplify to find the value of x:
x= 6.
Therefore, the solution to the proportion 2
3=4
xis x= 6.
Question 31
Question
Simplify the following expression involving ratios and proportions:
3
5∇ · 1 + 2
3
Solution
Step 1: We start by simplifying the expression within the parentheses. Step 2:
We add 1 and 2
3. Step 3: This gives us 1 + 2
3=3
3+2
3=5
3. Step 4: Next, we
rewrite the expression with the simplified parentheses:
3
5∇ · 5
3
Step 5: To divide fractions, we multiply by the reciprocal of the denominator.
Step 6: This gives us 3
5×3
5. Step 7: We multiply the numerators together
and the denominators together. Step 8: So, 3×3
5×5=9
25 . Step 9: Therefore, the
simplified expression is 9
25 .
Question 32
Question
If 4 workers can finish a job in 6 days, how many workers are needed to finish
the same job in 3 days?
22
Solution
Step 1: Let’s first calculate the total work required to finish the job. Given that
4 workers can finish the job in 6 days, we can find the amount of work done by
each worker in 1 day. Let xbe the amount of work done by each worker in 1
day. So the total work required to finish the job is 4 ×xin 6 days.
Total work = 4x×6 = 24x
Step 2: Now, we need to find out how many workers are needed to finish the
job in 3 days. Let ybe the number of workers required to finish the job in 3
days. Since the total work remains the same, we can set up a proportion:
4x×6 = y×3
24x= 3y
Step 3: We know that x
y=1
3. So, to find the number of workers required,
we need to solve the equation x
y=1
3for y.
x
y=1
3
3x=y
Therefore, to finish the same job in 3 days, we would need 3 workers.
Question 33
Question
If 3 tablespoons of sugar are needed to make 12 cookies, how many tablespoons
of sugar are needed to make 72 cookies?
Solution
Step 1: Let xrepresent the number of tablespoons of sugar needed to make 72
cookies. We can set up a proportion using the given information:
3
12 =x
72
Step 2: Solve for xby cross multiplying:
3×72 = 12 ×x
216 = 12x
Step 3: Divide both sides by 12 to solve for x:
x=216
12 = 18
Therefore, 18 tablespoons of sugar are needed to make 72 cookies.
23
Question 34
Question
Solve the following proportion for x:
2
x+ 3 =x+ 1
4
Solution
Step 1: Cross multiply to eliminate the fractions.
2·4=(x+ 3) ·(x+ 1)
8 = x2+ 4x+ 3
Step 2: Rearrange the equation into standard form.
x2+ 4x+ 3 −8=0
x2+ 4x−5 = 0
Step 3: Factor the quadratic equation.
x2+ 4x−5=(x+ 5)(x−1) = 0
Step 4: Set each factor to zero and solve for x.
x+ 5 = 0 (Case 1)
x=−5
x−1 = 0 (Case 2)
x= 1
Final Answer: x=−5 or x= 1.
Question 35
Question
A bag contains some red balls and blue balls. The ratio of red balls to blue balls
is 2:3. If the total number of balls in the bag is 100, how many of them are red?
Solution
Step 1: Let’s set up equations based on the given information. Let xrepresent
the number of red balls and yrepresent the number of blue balls. We have the
following:
ratio of red balls to blue balls = 2 : 3
total number of balls = 100
24
Step 2: From the ratio, we can write the following equation:
x
y=2
3
Step 3: We can also write an equation based on the total number of balls:
x+y= 100
Step 4: To solve these two equations simultaneously, we can first rewrite the
ratio equation as:
3x= 2y
Step 5: Now, we can use substitution or elimination to find the values of x
and y. Let’s use elimination. Multiply the second equation by 3 and add it to
the third equation:
(3x= 2y
3(x+y) = 300
Step 6: Simplify the equations:
(3x= 2y
3x+ 3y= 300
Step 7: Subtract the first equation from the second equation:
3y−2y= 300
y= 100
Step 8: Now that we have found the number of blue balls, we can find the
number of red balls by substituting back into the third equation:
x+ 100 = 100
x= 0
Step 9: Therefore, there are 0 red balls in the bag.
25
Solution
Step 1: Determine the ratio of cups of flour to cookies in the original recipe.
Let xrepresent the number of cups of flour needed to make 36 cookies. We can
set up a proportion based on the given information:
4 cups of flour
24 cookies =xcups of flour
36 cookies
Step 2: Solve for xby cross-multiplying.
4×36 = 24x
144 = 24x
Step 3: Solve for xto find the number of cups of flour needed.
x=144
24 = 6
Therefore, you will need 6 cups of flour to make 36 cookies.
Question 3
Question
If 3x+ 2y= 12, find the ratio of xto ywhen x= 3y.
Solution
Let’s first substitute x= 3yinto the equation 3x+ 2y= 12 and solve for y.
Step 1: Substitute x= 3yinto the equation.
3(3y)+2y= 12
9y+ 2y= 12
11y= 12
y=12
11
Step 2: Find the value of xusing x= 3y.
x= 3(12
11) = 36
11
Step 3: Calculate the ratio of xto y.
x
y=
36
11
12
11
=36
12 = 3
So, the ratio of xto ywhen x= 3yis 3 .
2
Question 4
Question
A recipe for a fruit punch requires mixing 3 cups of orange juice, 4 cups of
pineapple juice, and 5 cups of cranberry juice. If a caterer wants to make a
smaller batch using 1 cup of orange juice, how many cups of pineapple juice and
cranberry juice should be used to maintain the same ratio of ingredients?
Solution
Let xrepresent the number of cups of pineapple juice needed, and yrepresent
the number of cups of cranberry juice needed to maintain the same ratio of
ingredients in the smaller batch.
Step 1: Set up the proportion using the ratios of the original recipe and
the smaller batch: 3
1=4
x=5
y
Step 2: Solve the first proportion for x:
3x= 4 ⇒x=4
3
Step 3: Solve the second proportion for y:
5y= 3 ⇒y=3
5
Step 4: Now that we have the ratios for the smaller batch, determine the
number of cups needed for pineapple juice and cranberry juice:
Pineapple juice = 4
3cups
Cranberry juice = 3
5cups
Therefore, to make a smaller batch with 1 cup of orange juice, the caterer
would need 4
3cups of pineapple juice and 3
5cups of cranberry juice.
Question 5
Question
In a certain class, the ratio of the number of male students to the number of
female students is 3:5. If there are 120 students in total, how many female
students are there in the class?
3
Solution
Let xrepresent the number of male students and yrepresent the number of
female students.
Step 1: Write the given information as a ratio. The ratio of male students
to female students is 3:5, so we have x:y= 3 : 5.
Step 2: Write an equation based on the total number of students. Since
there are 120 students in total, we have x+y= 120.
Step 3: Solve the system of equations. From the ratio x:y= 3 : 5, we can
write x=3
5y.
Substitute this expression for xinto the equation x+y= 120:
3
5y+y= 120
3
5y+5
5y= 120
8
5y= 120
8y= 120 ×5
8y= 600
y=600
8
y= 75
Step 4: Find the number of female students. There are 75 female students
in the class.
Question 6
Question
Solve the following proportion for x:
1
2x+ 3 =3x
4
4
Solution
To solve the given proportion, we can cross multiply to eliminate the fractions.
Step 1: Cross multiply to obtain:
4·1 = (2x+ 3) ·3x
Step 2: Simplify both sides of the equation:
4=6x2+ 9x
Step 3: Rearrange the equation to set it equal to zero:
6x2+ 9x−4=0
Step 4: To solve the quadratic equation, we can use the quadratic formula:
x=−b±√b2−4ac
2a
where a= 6, b= 9, and c=−4.
Step 5: Substitute the values of a,b, and cinto the quadratic formula and
solve for x:
x=−9±p92−4(6)(−4)
2(6)
x=−9±√81 + 96
12
x=−9±√177
12
Step 6: Therefore, the solutions for xare:
x=−9 + √177
12 or x=−9−√177
12
Question 7
Question
Solve for x:2
x+3 =5
2x−1.
5
Solution
Step 1: Cross-multiply to get rid of the fractions.
2
x+ 3 =5
2x−1
2(2x−1) = 5(x+ 3)
Step 2: Expand both sides of the equation.
4x−2=5x+ 15
Step 3: Rearrange the equation to isolate xon one side.
4x−5x= 15 + 2
−x= 17
Step 4: Solve for xby multiplying both sides by −1.
x=−17
Therefore, the solution to the equation is x=−17.
Question 8
Question
If the ratio of xto yis 3:5 and the ratio of yto zis 4:7, find the ratio of xto z.
Solution
Given: Ratio of xto y:x:y= 3 : 5
Ratio of yto z:y:z= 4 : 7
To find the ratio of xto z, we need to find a relation between xand zusing
the given information about x,y, and z.
Step 1: From the ratio of xto y, we can express yin terms of xas follows:
y=5
3x
Step 2: From the ratio of yto z, we can express yin terms of zas follows:
y=4
7z
Step 3: Equating the two expressions for y, we have:
5
3x=4
7z
6
Step 4: To find the ratio of xto z, solve for zin terms of x:
z=7
4×5
3x
Simplifying, we get:
z=35
12x
Step 5: Therefore, the ratio of xto zis x:z= 1 : 35
12 . Simplifying, we get
x:z= 12 : 35.
Thus, the ratio of xto zis 12:35.
Question 9
Question
If a mixture of paint is made by mixing red paint and blue paint in the ratio
3:5, how many liters of red paint should be mixed with 20 liters of blue paint
to make a mixture with a 40 liter ratio of red paint to blue paint?
Solution
Step 1: Find the amount of blue paint needed in the final mixture. Let xbe
the amount of red paint needed. Step 2: Write the ratio of red paint to blue
paint in the final mixture. Step 3: Set up the proportion using the ratios of red
paint to blue paint in the initial mixture and final mixture. Step 4: Solve the
proportion to find the amount of red paint needed.
Step 1:
Let xbe the amount of red paint needed in the final mixture.
Step 2:
The ratio of red paint to blue paint in the final mixture is 40:20 = 2:1.
Step 3:
Using the given ratios, we set up the proportion as follows:
3
5=x
20
2
1=x
20
7
Step 4:
Solving the proportion, we find:
3
5=x
20
3×20 = 5x
60 = 5x
x=60
5
x= 12
Therefore, 12 liters of red paint should be mixed with 20 liters of blue paint
to make a mixture with a 40 liter ratio of red paint to blue paint.
Question 10
Question
If 3 liters of water is mixed with 5 liters of alcohol, what is the ratio of water
to alcohol in the resulting mixture?
Solution
Step 1: Calculate the total volume of the resulting mixture.
Total volume = 3 liters + 5 liters = 8 liters
Step 2: Find the ratio of water to alcohol in the mixture. The ratio of water
to alcohol can be expressed as x:y. In this case, we will use 3xto represent the
volume of water and 5xto represent the volume of alcohol in the mixture.
3x+ 5x= 8
8x= 8
x= 1
Step 3: Therefore, the ratio of water to alcohol in the resulting mixture is:
3:5
8
Question 11
Question
Solve the proportion: 2x−5
3x+ 7 =8
11
Solution
Step 1: Cross multiply to eliminate the fractions:
(2x−5) ·11 = 8 ·(3x+ 7)
22x−55 = 24x+ 56
Step 2: Simplify the equation by collecting like terms:
22x−24x= 56 + 55
−2x= 111
Step 3: Divide by -2 to solve for x:
x=111
−2=−111
2
Therefore, the solution to the proportion is x=−111
2.
Question 12
Question
Solve the following proportion for x:
2x−3
x+ 5 =x+ 7
3x−2
Solution
Step 1: Cross multiply to get rid of the fractions.
(2x−3)(3x−2) = (x+ 5)(x+ 7)
6x2−4x−9x+ 6 = x2+ 7x+ 5x+ 35
6x2−13x+ 6 = x2+ 12x+ 35
Step 2: Simplify the equation and set it equal to zero.
6x2−13x+ 6 = x2+ 12x+ 35
5x2−25x−29 = 0
9
Step 3: Factor the quadratic equation.
5x2−25x−29 = 0
5(x2−5x)−29 = 0
5(x2−5x+25
4)−29 −5(25
4) = 0
5(x−5
2)2−29 −125
4= 0
5(x−5
2)2−181
4= 0
5(x−5
2)2=181
4
(x−5
2)2=181
20
Step 4: Solve for x.
x−5
2=±r181
20
x=5
2±r181
20
x=5±√181
2√5
x=5±√181
2√5·√5
√5
x=5√5±√905
10
Thus, the solutions for xare x=5√5+√905
10 and x=5√5−√905
10 .
Question 13
Question
If 3 men or 4 women can do a piece of work in 43 days. How long will 7 men
and 5 women together take to do the same work?
Solution
Step 1: Let’s assume that the work is represented by 1 unit. Thus, the work
rate of a man is 1
3×43 and the work rate of a woman is 1
4×43 .
Step 2: Next, we calculate the total work rate when 7 men and 5 women
work together:
71
3×43+ 5 1
4×43
10
Step 3: Simplifying the above expression, we get:
7
129 +5
172 =68
516 +45
516 =113
516
Step 4: Finally, we find the number of days required for 7 men and 5 women
to complete the work together by taking the reciprocal of the total work rate:
516
113 = 4 64
113 days
Therefore, 7 men and 5 women together will take 4 64
113 days to complete the
work.
Question 14
Question
A survey of 300 students showed that 45 of them like to play basketball. If
we assume that this sample is representative of the entire student population,
how many students in a school of 1500 students can we expect to like playing
basketball?
Solution
Step 1: Find the proportion of students who like to play basketball in the
sample. Step 2: Use the proportion to estimate the total number of students
who like playing basketball in the school of 1500 students.
Step 1: Calculate the proportion of students who like to play basketball in
the sample.
Proportion = Number of students who like basketball
Total number of students in the sample
Proportion = 45
300 =3
20
Step 2: Estimate the total number of students who like playing basketball
in the school of 1500 students.
Estimated number of students who like basketball = Proportion×Total number of students in the school
Estimated number of students who like basketball = 3
20 ×1500 = 225
Therefore, we can expect approximately 225 students in a school of 1500
students to like playing basketball.
11
Question 15
Question
If x,y, and zare three positive numbers such that x
y=3
2and y
z=5
4, find the
value of x+z
y.
Solution
Step 1: From the given ratios, we can set up the following equations:
x
y=3
2
y
z=5
4
Step 2: Rearranging the first equation, we have x=3
2y.
Step 3: Rearranging the second equation, we have z=4
5y.
Step 4: Substituting the expressions for xand zinto x+z
y, we get:
x+z
y=3
2y+4
5y
y
Step 5: Simplifying the expression inside the brackets, we have:
3
2y+4
5y
y=15
10 y+8
10 y
y=
23
10 y
y=23
10
Step 6: Therefore, the value of x+z
yis 23
10 .
Question 16
Question
Solve the proportion: 2x+8
3=x
6.
Solution
Step 1: Cross multiply to eliminate the fractions:
(2x+ 8) ×6=3×x
Step 2: Simplify both sides of the equation:
12x+ 48 = 3x
Step 3: Move all terms containing xto one side of the equation:
12x−3x=−48
12
Step 4: Simplify the left side of the equation:
9x=−48
Step 5: Solve for xby dividing both sides by 9:
x=−48
9=−16
3
Therefore, the solution to the proportion is x=−16
3.
Question 17
Question
If 12 workers can complete a project in 18 days, and 4 workers can complete
the same project in 54 days, how many more days would it take 8 workers to
complete the project compared to 12 workers?
Solution
Step 1: First, we find the work rate of each worker for both scenarios. Let r
be the work rate of each worker for the first scenario (12 workers in 18 days),
and r2be the work rate of each worker for the second scenario (4 workers in 54
days).
Step 2: For the first scenario, we have 12r×18 = 1 project, and for the
second scenario, we have 4r2×54 = 1 project. So, 12r=1
18 and 4r2=1
54 .
Step 3: Solve for rand r2.r=1
12×18 =1
216 and r2=1
4×54 =1
216 .
Step 4: Now, we find how long it takes 8 workers to complete the project.
Let dbe the number of days it takes 8 workers to complete the project.
Step 5: For 8 workers, we have 8 ×1
216 ×d= 1 project. Solving for d, we
get d=216
8.
Step 6: Calculate the number of days it takes 8 workers to complete the
project. d= 27 days.
Step 7: Finally, calculate the difference in days between 12 workers and 8
workers to complete the project. The difference is 27 −18 = 9 days. Therefore,
it would take 9 more days for 8 workers to complete the project compared to
12 workers.
Question 18
Question
If four quantities are in proportion, then adding 3 to the first and subtracting
2 from the second we get a new proportion. The terms in the new proportion
are 8, 9, 6, and 12. Find the original four quantities.
13
Solution
Let the original four quantities be a,b,c, and d, such that a
b=c
d. We are given
that after the transformation, the new proportion becomes a+3
b−2=c+3
d−2=8
6=
9
12 .
Step 1: Solve for aand busing the given new proportion.
Since a+3
b−2=8
6, we have:
a+ 3
b−2=8
6
a+ 3
b−2=4
3
3(a+ 3) = 4(b−2)
3a+ 9 = 4b−8
3a−4b=−17
Step 2: Solve for cand dusing the given new proportion.
Since c+3
d−2=9
12 , we have:
c+ 3
d−2=9
12
c+ 3
d−2=3
4
4(c+ 3) = 3(d−2)
4c+ 12 = 3d−6
4c−3d=−18
Step 3: Solve the system of equations to find the original quantities.
We need to solve the system of equations:
(3a−4b=−17
4c−3d=−18
Solving this system will give us the values of a,b,c, and d, which represent
the original four quantities.
Question 19
Question
Solve the following proportion for x:
3x
2=5
4−x
2
14
Solution
We will first simplify the proportion given and then solve for x.
Step 1: Simplify the proportion
3x
2=5
4−x
2
3x
2=5·2
4·2−x
2
3x
2=10
8−x
2
3x
2=5
4−x
2
3x
2=5−2x
4
Step 2: Cross multiply to solve for x
4·3x= 2 ·(5 −2x)
12x= 10 −4x
Step 3: Rearrange the equation to solve for x
12x+ 4x= 10
16x= 10
x=10
16
x=5
8
Therefore, the solution to the proportion is x=5
8.
Question 20
Question
Solve the proportion 2x+ 5
3x−4=7
8.
Solution
Step 1: Cross-multiply to obtain (2x+ 5) ·8 = 7 ·(3x−4). Step 2: Simplify
both sides to get 16x+40 = 21x−28. Step 3: Rearrange the equation to isolate
the variable by subtracting 16xfrom both sides and adding 28 to both sides:
40 + 28 = 21x−16x. Step 4: Simplify both sides to get 68 = 5x. Step 5:
Divide both sides by 5 to solve for x:x=68
5. Step 6: Thus, the solution to the
proportion is x=68
5.
15
Question 21
Question
If a car travels 180 miles in 3 hours, and another car travels 225 miles in 4 hours,
what is the ratio of their speeds?
Solution
Let’s denote the speed of the first car as s1and the speed of the second car as
s2. We can set up the following proportions based on the distances and times
given: 180 miles
3 hours =s1miles
1 hour and 225 miles
4 hours =s2miles
1 hour
From the first proportion, we find that s1= 60 mph, and from the second
proportion, we find that s2= 56.25 mph.
To find the ratio of their speeds, we divide the speed of the first car by the
speed of the second car:
s1
s2
=60 mph
56.25 mph =60
56.25 =80
75 =16
15
Therefore, the ratio of their speeds is 16 : 15 .
Question 22
Question
Solve for x:
4x−7
5x+ 9 =3x+ 2
2x−3
Solution
Step 1: Cross multiply to eliminate the fractions.
4x−7
5x+ 9 =3x+ 2
2x−3
⇒(4x−7)(2x−3) = (3x+ 2)(5x+ 9)
Step 2: Expand both sides of the equation.
8x2−12x−14x+ 21 = 15x2+ 27x+ 10x+ 18
⇒8x2−26x+ 21 = 15x2+ 37x+ 18
16
Step 3: Move terms to one side of the equation.
15x2+ 37x+ 18 −8x2+ 26x−21 = 0
7x2+ 63x−3=0
Step 4: Factor the quadratic equation.
(7x−1)(x+ 3) = 0
Step 5: Set each factor to zero to find the possible values of x.
7x−1 = 0 or x+ 3 = 0
⇒x=1
7or x=−3
Thus, the solutions are x=1
7or x=−3.
Question 23
Question
If 3 times the first of three consecutive odd integers is added to 4 times the
second, the result is 11 more than 5 times the third. Find the integers.
Solution
Let’s denote the three consecutive odd integers as x,x+ 2, and x+ 4.
Step 1: Set up the equation based on the given information. We
are given that 3x+ 4(x+ 2) = 5(x+ 4) + 11. We can simplify this equation to
solve for x.
Step 2: Solve for x.Expanding and simplifying the equation:
3x+ 4x+ 8 = 5x+ 20 + 11
7x+ 8 = 5x+ 31
7x−5x= 31 −8
2x= 23
x=23
2
x= 11.5
Since xshould be an integer, we made a mistake. Hence, there is no solution to
this problem.
17
Question 24
Question
A recipe for a cake calls for 3 cups of flour for every 1 cup of sugar. If you want
to make a cake using 9 cups of flour, how many cups of sugar should you use?
Solution
Step 1: Let’s set up a proportion using the information given in the question.
Let xbe the number of cups of sugar needed. We can set up the proportion:
3
1=9
x.
Step 2: Solve the proportion for x. Cross multiplying, we get 3x= 9. Divide
both sides by 3 to solve for x:x=9
3= 3.
Step 3: Therefore, you should use 3 cups of sugar when making the cake to
go with 9 cups of flour in the recipe.
Question 25
Question
A recipe for muffins calls for 2 cups of flour and 1 cup of sugar. If you want to
make 18 muffins instead of the original 12, how much flour should you use?
Solution
Step 1: First, let’s find the ratio of flour to sugar in the original recipe. - The
ratio of flour to sugar in the original recipe is 2 : 1.
Step 2: Next, we need to find the ratio of flour to sugar required to make 18
muffins. - Since we are increasing the number of muffins from 12 to 18, the new
ratio of flour to sugar would be x: 1, where xis the amount of flour needed for
18 muffins.
Step 3: Set up a proportion using the ratios from the original recipe and the
desired recipe for 18 muffins. 2
1=x
1
Step 4: Solve for x.
2 = x⇒x= 2
Step 5: Therefore, to make 18 muffins, you will need 2 cups of flour.
Question 26
Question
Solve the following proportion for x:
18
2x+ 3
4x−1=5
7
Solution
Step 1: Cross multiply to eliminate the fractions.
(2x+ 3) ·7=5·(4x−1)
Step 2: Expand both sides of the equation.
14x+ 21 = 20x−5
Step 3: Rearrange the equation by moving all terms involving xto one side.
14x+ 21 = 20x−5
14x−20x=−5−21
Step 4: Combine like terms.
−6x=−26
Step 5: Solve for xby dividing both sides by −6.
x=−26
−6
Step 6: Simplify the fraction to find the final answer.
x=13
3
Question 27
Question
If 2x+ 3y= 12 and x
2=y
3, find the values of xand y.
Solution
Step 1: Solve the second equation for xin terms of y.
x
2=y
3=⇒x=2y
3
Step 2: Substitute the expression for xinto the first equation and solve for
y.
22y
3+ 3y= 12
19
4y
3+ 3y= 12
4y+ 9y
3= 12
13y
3= 12
13y= 36 =⇒y=36
13
Step 3: Substitute the value of yback into the expression for xto find its
value.
x=2(36
13 )
3=72
39 =24
13
Therefore, the values of xand yare x=24
13 and y=36
13 .
Question 28
Question
If x:y= 4 : 3 and y:z= 5 : 6, find x:y:z.
Solution
Step 1: Let’s find a common ratio that relates x,y, and zby considering the
given ratios.
Step 2: Since x:y= 4 : 3 and y:z= 5 : 6, we can rewrite these ratios using
a common term. We do this by setting up a proportionality equation:
x
y=4
3and y
z=5
6
Step 3: By cross-multiplying in both equations, we get:
3x= 4yand 5y= 6z
Step 4: Solving for yin terms of xand z:
y=3
4xand y=6
5z
Step 5: Equating these two expressions for y, we have:
3
4x=6
5z
Step 6: Simplifying this equation to solve for zin terms of x:
z=5
4x
Step 7: Now we have the ratios x:y= 4 : 3 and y:z= 5 : 6, which can
be expressed as x:y:z= 20 : 15 : 18. Thus, the ratios of x,y, and zare
20 : 15 : 18.
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Question 29
Question
Solve for x:3x−10
2x+5 =x+2
3x−6.
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x−10)(3x−6) = (2x+ 5)(x+ 2)
9x2−18x−30x+ 60 = 2x2+ 5x+ 4x+ 10
9x2−48x+ 60 = 2x2+ 9x+ 10
Step 2: Combine like terms and move all terms to one side of the equation.
9x2−48x+ 60 −2x2−9x−10 = 0
7x2−57x+ 50 = 0
Step 3: Factor the quadratic equation.
(7x−5)(x−10) = 0
Step 4: Set each factor to zero and solve for x.
7x−5 = 0 (or) x−10 = 0
7x= 5 (or) x= 10
x=5
7(or) x= 10
Therefore, the solutions are x=5
7or x= 10.
Question 30
Question
Solve the following proportion for x:
2
3=4
x.
Solution
To solve the proportion 2
3=4
xfor x, we can cross multiply.
Step 1: Cross multiply to get:
2·x= 3 ·4.
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Step 2: Simplify both sides of the equation:
2x= 12.
Step 3: To solve for x, divide both sides by 2:
2x
2=12
2.
Step 4: Simplify to find the value of x:
x= 6.
Therefore, the solution to the proportion 2
3=4
xis x= 6.
Question 31
Question
Simplify the following expression involving ratios and proportions:
3
5∇ · 1 + 2
3
Solution
Step 1: We start by simplifying the expression within the parentheses. Step 2:
We add 1 and 2
3. Step 3: This gives us 1 + 2
3=3
3+2
3=5
3. Step 4: Next, we
rewrite the expression with the simplified parentheses:
3
5∇ · 5
3
Step 5: To divide fractions, we multiply by the reciprocal of the denominator.
Step 6: This gives us 3
5×3
5. Step 7: We multiply the numerators together
and the denominators together. Step 8: So, 3×3
5×5=9
25 . Step 9: Therefore, the
simplified expression is 9
25 .
Question 32
Question
If 4 workers can finish a job in 6 days, how many workers are needed to finish
the same job in 3 days?
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Solution
Step 1: Let’s first calculate the total work required to finish the job. Given that
4 workers can finish the job in 6 days, we can find the amount of work done by
each worker in 1 day. Let xbe the amount of work done by each worker in 1
day. So the total work required to finish the job is 4 ×xin 6 days.
Total work = 4x×6 = 24x
Step 2: Now, we need to find out how many workers are needed to finish the
job in 3 days. Let ybe the number of workers required to finish the job in 3
days. Since the total work remains the same, we can set up a proportion:
4x×6 = y×3
24x= 3y
Step 3: We know that x
y=1
3. So, to find the number of workers required,
we need to solve the equation x
y=1
3for y.
x
y=1
3
3x=y
Therefore, to finish the same job in 3 days, we would need 3 workers.
Question 33
Question
If 3 tablespoons of sugar are needed to make 12 cookies, how many tablespoons
of sugar are needed to make 72 cookies?
Solution
Step 1: Let xrepresent the number of tablespoons of sugar needed to make 72
cookies. We can set up a proportion using the given information:
3
12 =x
72
Step 2: Solve for xby cross multiplying:
3×72 = 12 ×x
216 = 12x
Step 3: Divide both sides by 12 to solve for x:
x=216
12 = 18
Therefore, 18 tablespoons of sugar are needed to make 72 cookies.
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Question 34
Question
Solve the following proportion for x:
2
x+ 3 =x+ 1
4
Solution
Step 1: Cross multiply to eliminate the fractions.
2·4=(x+ 3) ·(x+ 1)
8 = x2+ 4x+ 3
Step 2: Rearrange the equation into standard form.
x2+ 4x+ 3 −8=0
x2+ 4x−5 = 0
Step 3: Factor the quadratic equation.
x2+ 4x−5=(x+ 5)(x−1) = 0
Step 4: Set each factor to zero and solve for x.
x+ 5 = 0 (Case 1)
x=−5
x−1 = 0 (Case 2)
x= 1
Final Answer: x=−5 or x= 1.
Question 35
Question
A bag contains some red balls and blue balls. The ratio of red balls to blue balls
is 2:3. If the total number of balls in the bag is 100, how many of them are red?
Solution
Step 1: Let’s set up equations based on the given information. Let xrepresent
the number of red balls and yrepresent the number of blue balls. We have the
following:
ratio of red balls to blue balls = 2 : 3
total number of balls = 100
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Step 2: From the ratio, we can write the following equation:
x
y=2
3
Step 3: We can also write an equation based on the total number of balls:
x+y= 100
Step 4: To solve these two equations simultaneously, we can first rewrite the
ratio equation as:
3x= 2y
Step 5: Now, we can use substitution or elimination to find the values of x
and y. Let’s use elimination. Multiply the second equation by 3 and add it to
the third equation:
(3x= 2y
3(x+y) = 300
Step 6: Simplify the equations:
(3x= 2y
3x+ 3y= 300
Step 7: Subtract the first equation from the second equation:
3y−2y= 300
y= 100
Step 8: Now that we have found the number of blue balls, we can find the
number of red balls by substituting back into the third equation:
x+ 100 = 100
x= 0
Step 9: Therefore, there are 0 red balls in the bag.
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