MATH 114 - QUANTITATIVE
REASONING - Ratios and Proportions
Question Bank - Set 1
Liberty University
Question 1
Question
Solve for x:2x+ 3
5x−1=7
9.
Solution
Step 1: Cross multiply to get rid of the fractions:
(2x+ 3) ·9=7·(5x−1)
18x+ 27 = 35x−7
Step 2: Rearrange the equation to isolate the variable x:
18x+ 27 = 35x−7
27 + 7 = 35x−18x
35 = 17x
Step 3: Solve for x:
17x= 35
x=35
17
Question 2
Question
Solve the following ratio and proportion problem:
If xand yare two numbers such that the ratio of xto yis 3 : 4, and the
sum of 2xand 3yis 70, find the values of xand y.
Solution
Let’s begin solving the problem step by step:
Step 1: Write down the given ratio. Given that the ratio of xto yis 3 : 4,
we can express this as x
y=3
4.
Step 2: Set up the equation using the sum of 2xand 3y. Since the sum of
2xand 3yis 70, we can write this as: 2x+ 3y= 70.
Step 3: Use the ratio to express one variable in terms of the other. From
the ratio x
y=3
4, we can express xin terms of yas x=3
4y.
Step 4: Substitute the expression for xinto the equation from Step 2. Sub-
stituting x=3
4yinto 2x+ 3y= 70, we get: 2(3
4y)+3y= 70.
Step 5: Solve for y. Solving the equation gives: 6
4y+ 3y= 70
6y
4+ 3y= 70
6y+12y
4= 70
18y
4= 70
18y= 280
y=280
18
y= 15.56 (rounded to two decimal places).
Step 6: Find the value of xusing the ratio. Substitute y= 15.56 into x=3
4y
to find x:x=3
4×15.56
x= 11.67 (rounded to two decimal places).
Hence, the values of xand yare x= 11.67 and y= 15.56, respectively.
Question 3
Question
Given three numbers a,b, and csuch that a:b= 2 : 3 and b:c= 4 : 5, find
the ratio of ato c.
Solution
Let’s first express a,b, and cin terms of a common variable using the given
ratios.
Step 1: Express aand bin terms of k: Since a:b= 2 : 3, we can express a
and bas: a= 2kand b= 3k.
Step 2: Express band cin terms of m: Since b:c= 4 : 5, we can rewrite b
and cas: b= 4mand c= 5m.
Step 3: Find the ratio of ato c: Substitute the expressions for aand cin
terms of kand mrespectively: a
c=2k
5m.
Hence, the ratio of ato cis 2 : 5 .
2
Question 4
Question
Solve for xin the following proportion: 3x+2
x−4=5
2.
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x+ 2) ·2 = 5 ·(x−4)
6x+ 4 = 5x−20
Step 2: Subtract 5xfrom both sides.
6x+ 4 −5x= 5x−20 −5x
x+ 4 = −20
Step 3: Subtract 4 from both sides.
x+ 4 −4 = −20 −4
x=−24
Therefore, the solution is x=−24.
Question 5
Question
A construction company is building a bridge that is 200 meters long. The
company has calculated that it will take 10 days to complete the bridge if 20
workers are employed. If the company wants to finish the bridge in 6 days, how
many additional workers should they hire?
Solution
Step 1: Calculate the current rate at which the workers are completing the
bridge. Let Rbe the rate at which one worker completes the bridge in 10 days.
The total work to be completed is building a 200-meter bridge. Therefore, the
current rate of completion is given by:
Total work = R×20 ×10 = 200
Step 2: Solve for R.
R=200
200 = 1 meter/day
3
Step 3: Calculate the new rate needed to finish the bridge in 6 days. Let x
be the number of additional workers needed to complete the bridge in 6 days.
The new total work to be completed is still building a 200-meter bridge. The
new rate of completion when 20 + xworkers are employed is given by:
Total work = R×(20 + x)×6 = 200
Step 4: Solve for x.
1×(20 + x)×6 = 200
120 + 6x= 200
6x= 80
x=80
6= 13.3
Therefore, the construction company should hire 14 additional workers to
finish the bridge in 6 days.
Question 6
Question
Solve the following proportion: 3x+4
6x−8=5
7.
Solution
To solve the proportion 3x+4
6x−8=5
7, we need to cross multiply and solve for x.
Step 1: Cross multiply the fractions to obtain (3x+ 4) ·7 = 5 ·(6x−8).
21x+ 28 = 30x−40
Step 2: Rearrange the equation by moving all terms involving xto one side
of the equation.
28 + 40 = 30x−21x
68 = 9x
Step 3: Solve for xby dividing both sides by 9.
x=68
9
Therefore, the solution to the proportion 3x+4
6x−8=5
7is x=68
9.
Question 7
Question
If x, y, and zare positive real numbers such that x:y= 2 : 3 and y:z= 3 : 4,
find the ratio x:z.
4
Solution
We are given that x:y= 2 : 3 and y:z= 3 : 4. We need to find x:z.
Step 1: Write the ratios in fraction form. Given x:y= 2 : 3, we can write
this as x
y=2
3. Similarly, y:z= 3 : 4 can be written as y
z=3
4.
Step 2: Set up an equation using the fractions. Since x
y=2
3and y
z=3
4, we
can write x
y·y
z=2
3·3
4.
Step 3: Simplify the equation. Multiplying the fractions gives us x
z=1
2.
Step 4: Write the ratio x:z. Therefore, the ratio x:zis 1 : 2 .
Question 8
Question
Solve for xin the proportion: 3x
2=4
5x.
Solution
Step 1: Cross multiply to eliminate the fractions.
3x
2=4
5x
3x·5x= 2 ·4
Step 2: Simplify the equation.
15x2= 8
Step 3: Divide by 15 to isolate x2.
x2=8
15
Step 4: Take the square root of both sides to solve for x.
x=±r8
15
Step 5: Simplify the square root.
x=±2√15
5
Therefore, the solutions for xare x=2√15
5and x=−2√15
5.
Question 9
Question
If xis directly proportional to y2and inversely proportional to z, and x= 6
when y= 3 and z= 4, find xwhen y= 5 and z= 6.
5
Solution
Step 1: Write the equation of proportionality using the given information. Since
xis directly proportional to y2and inversely proportional to z, we have:
x=ky2·1
z
where kis a constant of proportionality.
Step 2: Use the initial conditions x= 6, y= 3, and z= 4 to find k.
Substitute these values into the equation to get:
6 = k·32·1
4
6=9k·1
4
6 = 9
4k
k=24
9=8
3
Step 3: Substitute the value of kinto the equation of proportionality. We
now have:
x=8
3y2·1
z
Step 4: Use the new values of y= 5 and z= 6 to find x. Substitute these
values into the equation:
x=8
3·52·1
6
x=8
3·25 ·1
6
x=200
3·1
6
x=200
18
x=100
9= 11.1
Therefore, when y= 5 and z= 6, x≈11.1.
Question 10
Question
If 8 men can complete a job in 12 days, and 6 women can complete the same
job in 18 days, how many days will it take for 12 men and 9 women working
together to complete the job?
6
Solution
Step 1: Calculate the man-day ratio for completing the job. Let Mbe the
number of men required to finish the job in 1 day. Given that 8 men complete
the job in 12 days,
M=8 men ×12 days
1= 96 man-days
Step 2: Calculate the woman-day ratio for completing the job. Let Wbe
the number of women required to finish the job in 1 day. Given that 6 women
complete the job in 18 days,
W=6 women ×18 days
1= 108 woman-days
Step 3: Calculate the combined man-day and woman-day ratio for complet-
ing the job together. Let Dbe the number of days required for 12 men and 9
women to complete the job together. Using the man-day and woman-day ratios:
12M+ 9W=D
12(96) + 9(108) = D
D= 1152 + 972 = 2124 man-woman days
Step 4: Determine the number of days needed to complete the job when
working together. Since 12 men and 9 women are working together,
D=12 men + 9 women
1 day ×tdays
2124 = 12 ×96 + 9 ×108 ×t
2124 = 1152 + 972t
972t= 972
t= 1 day
Therefore, it will take 12 men and 9 women working together 1 day to
complete the job.
Question 11
Question
Solve the following proportion for x:
3
x+ 1 =x
4
7
Solution
To solve the given proportion, we will cross-multiply and then solve for x.
Step 1: Cross-multiply to get rid of the fractions:
3·4 = x·(x+ 1)
Step 2: Simplify both sides of the equation:
12 = x2+x
Step 3: Rearrange the equation into a quadratic form:
x2+x−12 = 0
Step 4: Factor the quadratic equation:
(x+ 4)(x−3) = 0
Step 5: Set each factor to zero to solve for x:
For x+ 4 = 0:
x=−4
For x−3 = 0:
x= 3
Step 6: Check for extraneous solutions:
Since both x=−4 and x= 3 satisfy the original proportion, there are no
extraneous solutions.
Therefore, the solutions to the given proportion are x=−4 and x= 3.
Question 12
Question
Solve the following proportion for x:
3x−1
x+ 7 =4x+ 5
2x−3
Solution
To solve the proportion 3x−1
x+7 =4x+5
2x−3, we can cross multiply.
Step 1: Cross multiply to get rid of the fractions:
(3x−1)(2x−3) = (x+ 7)(4x+ 5)
Expanding both sides gives:
6x2−9x−2x+ 3 = 4x2+ 5x+ 28x+ 35
8
Simplify to get:
6x2−11x+ 3 = 4x2+ 33x+ 35
Step 2: Rearrange the equation to solve for x: Subtract 4x2, 33x, and 35
from both sides:
2x2−44x−32 = 0
Step 3: Solve the quadratic equation: To factor the quadratic equation
2x2−44x−32 = 0, we simplify by dividing everything by 2 to get:
x2−22x−16 = 0
The equation can be factored as:
(x−24)(x+ 2) = 0
Setting each factor to zero gives:
x−24 = 0 or x+ 2 = 0
So the solutions are:
x= 24 or x=−2
Therefore, the solutions to the proportion 3x−1
x+7 =4x+5
2x−3are x= 24 and
x=−2.
Question 13
Question
Solve the proportion: 2x+3
x−1=x+4
x.
Solution
Step 1: Cross multiply to eliminate the fractions.
(x−1)(x+ 4) = (2x+ 3)(x)
x2+ 3x−4=2x2+ 3x
Step 2: Rearrange the equation to set it equal to zero.
0 = 2x2+ 3x−x2−3x−4
0 = x2−4
Step 3: Factor the quadratic equation.
0=(x+ 2)(x−2)
9
Step 4: Set each factor equal to zero and solve for x.
x+ 2 = 0 x−2=0
x=−2x= 2
Step 5: Check the solutions in the original proportion. Checking x=−2:
2(−2) + 3
−2−1=−2+4
−2
−1
−3=2
−2
1
3=−1
Step 6: Final answer. The solution to the proportion is x= 2.
Question 14
Question
In a certain bag of marbles, the ratio of blue marbles to red marbles is 3:5. If
there are a total of 64 marbles in the bag, how many red marbles are there?
Solution
Step 1: Let’s denote the number of blue marbles as 3xand the number of red
marbles as 5x, where xis a positive integer.
Step 2: We know that the total number of marbles is 64. Therefore, we have
the equation:
3x+ 5x= 64
Step 3: Simplifying the equation, we get:
8x= 64
Step 4: Dividing both sides of the equation by 8, we find:
x= 8
Step 5: Now, we can determine the number of red marbles by substituting
x= 8 back into our expression for the number of red marbles:
5x= 5(8) = 40
Step 6: Thus, there are 40 red marbles in the bag.
10
Question 15
Question
If xis directly proportional to yand inversely proportional to z, and x= 24
when y= 6 and z= 4, find the value of xwhen y= 8 and z= 6.
Solution
Let’s first express the proportionalities given:
xis directly proportional to y:x=ky for some constant k.
xis inversely proportional to z:x=k
zfor the same constant k.
Step 1: Find the value of kusing the first set of values given.
24 = k×6⇒k= 4
Step 2: Use the value of kto find the value of xwhen y= 8 and z= 6.
x= 4 ×8 = 32
Therefore, when y= 8 and z= 6, the value of xis 32.
Question 16
Question
Solve the following proportion for x:
3
x+ 2 =5
4x−1
Solution
To solve the proportion 3
x+2 =5
4x−1for x, we can cross-multiply to get rid of
the fractions.
Step 1: Cross-multiply to get:
3(4x−1) = 5(x+ 2)
Step 2: Expand both sides of the equation:
12x−3=5x+ 10
Step 3: Simplify the equation:
12x−5x= 10 + 3
11
7x= 13
Step 4: Divide both sides by 7 to solve for x:
x=13
7
Therefore, the solution to the proportion is x=13
7.
Question 17
Question
Solve the proportion: 4x
6=x+ 3
9
Solution
To solve the proportion, we will cross multiply and then solve for x.
Step 1: Cross multiply to get:
4x·9=6·(x+ 3)
Simplify both sides:
36x= 6x+ 18
Step 2: Move all terms involving xto one side of the equation:
36x−6x= 18
Simplify the left side of the equation:
30x= 18
Step 3: Solve for xby dividing both sides by 30:
x=18
30
Simplify the fraction:
x=3
5
Therefore, the solution to the proportion is x=3
5.
Question 18
Question
Solve the following proportion for x:
1
2=3
x+ 2
12
Solution
To solve the proportion given, we can cross multiply to eliminate the fractions
and then solve for x.
Step 1: Cross multiply the terms.
1(x+ 2) = 2 ·3
Step 2: Simplify both sides of the equation.
x+ 2 = 6
Step 3: Subtract 2 from both sides to solve for x.
x= 6 −2
Step 4: Calculate the value of x.
x= 4
Therefore, the value of xthat satisfies the proportion is 4.
Question 19
Question
Solve for xin the proportion 4
2x−1=x+ 3
5.
Solution
Step 1: Cross multiply to eliminate the fractions.
4·5 = (2x−1)(x+ 3)
20 = 2x2+ 6x−x−3
20 = 2x2+ 5x−3
Step 2: Rearrange the equation into standard quadratic form.
2x2+ 5x−3 = 20
2x2+ 5x−23 = 0
Step 3: Solve the quadratic equation using the quadratic formula: x=
−b±√b2−4ac
2a, where a= 2, b= 5, and c=−23.
x=−5±p52−4(2)(−23)
2(2)
x=−5±√25 + 184
4
x=−5±√209
4
13
Thus, the solutions for xare x=−5 + √209
4and x=−5−√209
4.
Question 20
Question
Solve the following proportion for x:3
x=5
12 .
Solution
Step 1: Cross multiply to solve for x.
3
x=5
12
3×12 = 5 ×x
36 = 5x
Step 2: Divide by 5 to solve for x.
x=36
5
x= 7.2
Therefore, the value of xis 7.2.
Question 21
Question
If 3x
2y=4y
5z, find x
zin terms of y.
Solution
Step 1: Multiply both sides of the equation by 2yz to clear the fractions.
3x
2y·2yz =4y
5z·2yz
⇒3xz = 8y
Step 2: Solve for x
zby isolating x
z.
x
z=8y
3
x
z=8y
3
14
Question 22
Question
Solve the proportion: 3x+4
5=8
x+2 .
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x+ 4)(x+ 2) = 5 ·8
3x2+ 10x+ 8 = 40
Step 2: Rearrange the equation and simplify.
3x2+ 10x+ 8 = 40
3x2+ 10x−32 = 0
Step 3: Factor the quadratic equation.
3x2+ 16x−6x−32 = 0
3x(x+ 5) −2(x+ 5) = 0
(3x−2)(x+ 5) = 0
Step 4: Solve for x. If 3x−2 = 0, then 3x= 2, x=2
3. If x+ 5 = 0, then
x=−5.
Therefore, the solutions to the proportion are x=2
3and x=−5.
Question 23
Question
Solve the following proportion for x:
7
10 =x
15
Solution
To solve the proportion 7
10 =x
15 for x, we can use the property that in a
proportion, the cross products are equal.
Step 1: Cross multiply to get:
7×15 = 10x
Step 2: Simplify the equation:
105 = 10x
15
Step 3: Divide by 10 to solve for x:
x=105
10 = 10.5
Therefore, the solution to the proportion is x= 10.5.
Question 24
Question
Solve the proportion: 3 + x
2=x+ 5
3.
Solution
Step 1: Cross-multiply to eliminate the fractions.
3(3 + x) = 2(x+ 5)
9+3x= 2x+ 10
Step 2: Move all terms involving xto one side of the equation.
3x−2x= 10 −9
x= 1
Step 3: Check the solution by substituting x= 1 back into the original
proportion. 3+1
2=1+5
3
4
2=6
3
2=2
Step 4: Therefore, the solution to the proportion is x= 1 .
Question 25
Question
If a:b= 5 : 8 and b:c= 2 : 3, find the ratio a:b:c.
Solution
Let us first find the value of bby setting up equations based on the given ratios.
16
Step 1: Find the value of b.
a
b=5
8
a=5
8b
b
c=2
3
c=3
2b
Step 2: Find the value of busing the two equations derived above.
a=5
8b(Substitute the value of a)
=5
83
2b
=15
16b
Therefore, b=16
15 a.
Step 3: Compile the ratio a:b:c.
a:b:c=a:16
15a:3
2·16
15a
= 15a: 16a: 32a
= 15 : 16 : 32
Question 26
Question
A recipe calls for 4 cups of flour to make 36 cookies. If you only have 1 cup of
flour, how many cookies can you make using the same ratio?
Solution
Let xrepresent the number of cookies that can be made using 1 cup of flour.
Step 1: Set up a proportion using the given information. By setting up a
proportion with the amount of flour and the number of cookies, we get:
4 cups of flour
36 cookies =1 cup of flour
xcookies
Step 2: Solve the proportion. Cross-multiplying, we get:
4·x= 36 ·1
17
4x= 36
x=36
4
x= 9
Step 3: Answer the question. Therefore, you can make 9 cookies using 1
cup of flour with the same ratio as in the original recipe.
Question 27
Question
Solve the following proportion for x:
3
4=2
x.
Solution
Step 1: Cross multiply to solve the proportion.
3·x= 4 ·2
3x= 8
Step 2: Divide both sides by 3 to solve for x.
3x
3=8
3
x=8
3
Therefore, the solution to the proportion 3
4=2
xis x=8
3.
Question 28
Question
If x:y= 3 : 7 and y:z= 5 : 4, find the ratio x:z.
Solution
Step 1: Let’s write the given ratios in fraction form:
x
y=3
7and y
z=5
4
Step 2: Since we want to find the ratio x:z, we need to find a relationship
between xand zin terms of y.
18
Step 3: From the given ratios, we can see that there is a common term y.
To eliminate y, we can equate the two expressions for y:
x
y=3
7=y
z=5
4
Step 4: Solving for yin terms of xand zfrom these equations, we get:
y=7x
3and y=5z
4
Step 5: Equating these two expressions for ygives us an equation involving
xand z:7x
3=5z
4
Step 6: Solving for x:zin terms of x, we find:
x
z=3
7×4
5=12
35
Step 7: Therefore, the ratio x:zis 12 : 35.
Question 29
Question
Solve the proportion: 3x−1
x+2 =5
4.
Solution
To solve the given proportion, we can cross multiply to get an equation in x.
Step 1: Cross multiply to get (3x−1) ·4=5·(x+ 2).
(3x−1) ·4 = 5 ·(x+ 2)
12x−4=5x+ 10
Step 2: Simplify the equation by combining like terms.
12x−4=5x+ 10
12x−5x= 10 + 4
7x= 14
Step 3: Solve for xby dividing both sides by 7.
7x= 14
7x
7=14
7
19
x= 2
Step 4: Check the solution by substituting x= 2 back into the original
proportion.
3(2) −1
2+2 =5
4
6−1
4=5
4
5
4=5
4
Since both sides of the proportion are equal with x= 2, our solution is
correct. Thus, x= 2 satisfies the given proportion.
Question 30
Question
Solve the following proportion for x:4x−1
2x+ 3 =5
6.
Solution
Step 1: Cross multiply to eliminate the fractions:
6(4x−1) = 5(2x+ 3)
24x−6 = 10x+ 15
Step 2: Simplify the equation by combining like terms:
24x−10x= 15 + 6
14x= 21
Step 3: Solve for xby dividing both sides by 14:
x=21
14
Step 4: Simplify the fraction to get the final answer:
x=3
2
Therefore, x=3
2.
20
Question 31
Question
A recipe for a certain dish requires 2 pounds of ingredients to produce 3 servings.
If you want to make 5 servings of the dish, how many pounds of ingredients will
you need?
Solution
Step 1: Calculate the ratio of pounds of ingredients to servings in the recipe.
Let the pounds of ingredients required be xfor 3 servings. So, the ratio of
pounds of ingredients to servings is 2/3 = x/3.
Step 2: Calculate the amount of ingredients required for 5 servings. Now we
need to find how many pounds of ingredients are needed for 5 servings. Using
the ratio from Step 1: 2/3 = x/3 Cross multiplying, we get: 3x= 2 ×3 3x= 6
x= 2 pounds
Therefore, to make 5 servings, you will need 3 pounds of ingredients.
Question 32
Question
A group of university students rented a van for a road trip. They agreed to split
the total cost equally. If 6 students bailed out at the last minute, each of the re-
maining students would need to pay 30more.Howmuchwasthetotalcostof rentingthevan?
Solution
Let xbe the total cost of renting the van and nbe the number of students who
initially agreed to split the cost. After 6 students bailed out, the number of
students left is n−6.
Step 1: Set up the equation based on the information provided:
x
n=x
n−6+ 30
Step 2: Clear the fractions by multiplying through by n(n−6):
x(n−6) = x·n+ 30n(n−6)
Step 3: Expand both sides of the equation:
xn −6x=xn + 30n2−180n
Step 4: Simplify the equation:
30n2−180n= 6x
21
5n2−30n=x
Step 5: Since the total cost is divided equally among the students, we have:
x=n·cost per student
Step 6: Substitute x= 5n2−30ninto x=n·cost per student:
5n2−30n=n·cost per student
Step 7: Solve for the cost per student:
cost per student = 5n2−30n
n= 5n−30
Step 8: From the given information, we know that after 6 students bailed out,
the remaining students had to pay 30more.T hisgivesustheequation : 5n−30 =
5n−30 + 30
Step 9: Solve for n:
5n−30 = 5n
−30 = 0
Step 10: This equation has no solution, which means our initial assumption
was incorrect. This can happen if the total cost of renting the van was not
properly divided among the students.
Question 33
Question
In a bakery, the ratio of the number of chocolate chip cookies to the number of
oatmeal raisin cookies is 4:7. If there are 72 oatmeal raisin cookies, how many
chocolate chip cookies are there?
Solution
Step 1: Determine the multiplier from the given ratio. Let the number of
chocolate chip cookies be represented by 4x, where x is the multiplier. Since
the ratio of chocolate chip cookies to oatmeal raisin cookies is 4:7, the number
of oatmeal raisin cookies is 7x.
Step 2: Use the information given in the problem to find the value of x. Given
that there are 72 oatmeal raisin cookies: 7x = 72 Solving for x: x=72
7= 102
7
Step 3: Calculate the number of chocolate chip cookies. Substitute the value
of x back into the expression for chocolate chip cookies: Number of chocolate
chip cookies = 4x Number of chocolate chip cookies = 4 ×102
7= 416
7
Therefore, there are 41 chocolate chip cookies in the bakery.
22
Question 34
Question
A bakery sells muffins and cookies in the ratio 3:4. If the bakery sells 210
muffins, how many cookies does it sell?
Solution
Let xbe the number of cookies sold by the bakery.
Step 1: Set up a proportion based on the ratio of muffins to cookies:
3
4=210
x
Step 2: Cross multiply to solve for x:
3·x= 4 ·210
Step 3: Simplify the equation:
3x= 840
Step 4: Divide both sides by 3 to solve for x:
x=840
3= 280
The bakery sells 280 cookies.
Question 35
Question
A group of 12 university students decided to share equally the cost of a trip.
Three students dropped out, and the remaining students had to increase their
contribution by $5 each to cover the cost. How much did the trip cost in total?
Solution
Step 1: Let’s denote the total cost of the trip as Tand each student’s initial
contribution as x. Since the cost is shared equally among 12 students:
12x=T
Step 2: After three students dropped out, there were only 9 students left.
Each student had to contribute an extra 5, so the total contribution per student
became x+ 5. The total cost was still T:
9(x+ 5) = T
23
Solution
Let’s begin solving the problem step by step:
Step 1: Write down the given ratio. Given that the ratio of xto yis 3 : 4,
we can express this as x
y=3
4.
Step 2: Set up the equation using the sum of 2xand 3y. Since the sum of
2xand 3yis 70, we can write this as: 2x+ 3y= 70.
Step 3: Use the ratio to express one variable in terms of the other. From
the ratio x
y=3
4, we can express xin terms of yas x=3
4y.
Step 4: Substitute the expression for xinto the equation from Step 2. Sub-
stituting x=3
4yinto 2x+ 3y= 70, we get: 2(3
4y)+3y= 70.
Step 5: Solve for y. Solving the equation gives: 6
4y+ 3y= 70
6y
4+ 3y= 70
6y+12y
4= 70
18y
4= 70
18y= 280
y=280
18
y= 15.56 (rounded to two decimal places).
Step 6: Find the value of xusing the ratio. Substitute y= 15.56 into x=3
4y
to find x:x=3
4×15.56
x= 11.67 (rounded to two decimal places).
Hence, the values of xand yare x= 11.67 and y= 15.56, respectively.
Question 3
Question
Given three numbers a,b, and csuch that a:b= 2 : 3 and b:c= 4 : 5, find
the ratio of ato c.
Solution
Let’s first express a,b, and cin terms of a common variable using the given
ratios.
Step 1: Express aand bin terms of k: Since a:b= 2 : 3, we can express a
and bas: a= 2kand b= 3k.
Step 2: Express band cin terms of m: Since b:c= 4 : 5, we can rewrite b
and cas: b= 4mand c= 5m.
Step 3: Find the ratio of ato c: Substitute the expressions for aand cin
terms of kand mrespectively: a
c=2k
5m.
Hence, the ratio of ato cis 2 : 5 .
2
Question 4
Question
Solve for xin the following proportion: 3x+2
x−4=5
2.
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x+ 2) ·2 = 5 ·(x−4)
6x+ 4 = 5x−20
Step 2: Subtract 5xfrom both sides.
6x+ 4 −5x= 5x−20 −5x
x+ 4 = −20
Step 3: Subtract 4 from both sides.
x+ 4 −4 = −20 −4
x=−24
Therefore, the solution is x=−24.
Question 5
Question
A construction company is building a bridge that is 200 meters long. The
company has calculated that it will take 10 days to complete the bridge if 20
workers are employed. If the company wants to finish the bridge in 6 days, how
many additional workers should they hire?
Solution
Step 1: Calculate the current rate at which the workers are completing the
bridge. Let Rbe the rate at which one worker completes the bridge in 10 days.
The total work to be completed is building a 200-meter bridge. Therefore, the
current rate of completion is given by:
Total work = R×20 ×10 = 200
Step 2: Solve for R.
R=200
200 = 1 meter/day
3
Step 3: Calculate the new rate needed to finish the bridge in 6 days. Let x
be the number of additional workers needed to complete the bridge in 6 days.
The new total work to be completed is still building a 200-meter bridge. The
new rate of completion when 20 + xworkers are employed is given by:
Total work = R×(20 + x)×6 = 200
Step 4: Solve for x.
1×(20 + x)×6 = 200
120 + 6x= 200
6x= 80
x=80
6= 13.3
Therefore, the construction company should hire 14 additional workers to
finish the bridge in 6 days.
Question 6
Question
Solve the following proportion: 3x+4
6x−8=5
7.
Solution
To solve the proportion 3x+4
6x−8=5
7, we need to cross multiply and solve for x.
Step 1: Cross multiply the fractions to obtain (3x+ 4) ·7 = 5 ·(6x−8).
21x+ 28 = 30x−40
Step 2: Rearrange the equation by moving all terms involving xto one side
of the equation.
28 + 40 = 30x−21x
68 = 9x
Step 3: Solve for xby dividing both sides by 9.
x=68
9
Therefore, the solution to the proportion 3x+4
6x−8=5
7is x=68
9.
Question 7
Question
If x, y, and zare positive real numbers such that x:y= 2 : 3 and y:z= 3 : 4,
find the ratio x:z.
4
Solution
We are given that x:y= 2 : 3 and y:z= 3 : 4. We need to find x:z.
Step 1: Write the ratios in fraction form. Given x:y= 2 : 3, we can write
this as x
y=2
3. Similarly, y:z= 3 : 4 can be written as y
z=3
4.
Step 2: Set up an equation using the fractions. Since x
y=2
3and y
z=3
4, we
can write x
y·y
z=2
3·3
4.
Step 3: Simplify the equation. Multiplying the fractions gives us x
z=1
2.
Step 4: Write the ratio x:z. Therefore, the ratio x:zis 1 : 2 .
Question 8
Question
Solve for xin the proportion: 3x
2=4
5x.
Solution
Step 1: Cross multiply to eliminate the fractions.
3x
2=4
5x
3x·5x= 2 ·4
Step 2: Simplify the equation.
15x2= 8
Step 3: Divide by 15 to isolate x2.
x2=8
15
Step 4: Take the square root of both sides to solve for x.
x=±r8
15
Step 5: Simplify the square root.
x=±2√15
5
Therefore, the solutions for xare x=2√15
5and x=−2√15
5.
Question 9
Question
If xis directly proportional to y2and inversely proportional to z, and x= 6
when y= 3 and z= 4, find xwhen y= 5 and z= 6.
5
Solution
Step 1: Write the equation of proportionality using the given information. Since
xis directly proportional to y2and inversely proportional to z, we have:
x=ky2·1
z
where kis a constant of proportionality.
Step 2: Use the initial conditions x= 6, y= 3, and z= 4 to find k.
Substitute these values into the equation to get:
6 = k·32·1
4
6=9k·1
4
6 = 9
4k
k=24
9=8
3
Step 3: Substitute the value of kinto the equation of proportionality. We
now have:
x=8
3y2·1
z
Step 4: Use the new values of y= 5 and z= 6 to find x. Substitute these
values into the equation:
x=8
3·52·1
6
x=8
3·25 ·1
6
x=200
3·1
6
x=200
18
x=100
9= 11.1
Therefore, when y= 5 and z= 6, x≈11.1.
Question 10
Question
If 8 men can complete a job in 12 days, and 6 women can complete the same
job in 18 days, how many days will it take for 12 men and 9 women working
together to complete the job?
6
Solution
Step 1: Calculate the man-day ratio for completing the job. Let Mbe the
number of men required to finish the job in 1 day. Given that 8 men complete
the job in 12 days,
M=8 men ×12 days
1= 96 man-days
Step 2: Calculate the woman-day ratio for completing the job. Let Wbe
the number of women required to finish the job in 1 day. Given that 6 women
complete the job in 18 days,
W=6 women ×18 days
1= 108 woman-days
Step 3: Calculate the combined man-day and woman-day ratio for complet-
ing the job together. Let Dbe the number of days required for 12 men and 9
women to complete the job together. Using the man-day and woman-day ratios:
12M+ 9W=D
12(96) + 9(108) = D
D= 1152 + 972 = 2124 man-woman days
Step 4: Determine the number of days needed to complete the job when
working together. Since 12 men and 9 women are working together,
D=12 men + 9 women
1 day ×tdays
2124 = 12 ×96 + 9 ×108 ×t
2124 = 1152 + 972t
972t= 972
t= 1 day
Therefore, it will take 12 men and 9 women working together 1 day to
complete the job.
Question 11
Question
Solve the following proportion for x:
3
x+ 1 =x
4
7
Solution
To solve the given proportion, we will cross-multiply and then solve for x.
Step 1: Cross-multiply to get rid of the fractions:
3·4 = x·(x+ 1)
Step 2: Simplify both sides of the equation:
12 = x2+x
Step 3: Rearrange the equation into a quadratic form:
x2+x−12 = 0
Step 4: Factor the quadratic equation:
(x+ 4)(x−3) = 0
Step 5: Set each factor to zero to solve for x:
For x+ 4 = 0:
x=−4
For x−3 = 0:
x= 3
Step 6: Check for extraneous solutions:
Since both x=−4 and x= 3 satisfy the original proportion, there are no
extraneous solutions.
Therefore, the solutions to the given proportion are x=−4 and x= 3.
Question 12
Question
Solve the following proportion for x:
3x−1
x+ 7 =4x+ 5
2x−3
Solution
To solve the proportion 3x−1
x+7 =4x+5
2x−3, we can cross multiply.
Step 1: Cross multiply to get rid of the fractions:
(3x−1)(2x−3) = (x+ 7)(4x+ 5)
Expanding both sides gives:
6x2−9x−2x+ 3 = 4x2+ 5x+ 28x+ 35
8
Simplify to get:
6x2−11x+ 3 = 4x2+ 33x+ 35
Step 2: Rearrange the equation to solve for x: Subtract 4x2, 33x, and 35
from both sides:
2x2−44x−32 = 0
Step 3: Solve the quadratic equation: To factor the quadratic equation
2x2−44x−32 = 0, we simplify by dividing everything by 2 to get:
x2−22x−16 = 0
The equation can be factored as:
(x−24)(x+ 2) = 0
Setting each factor to zero gives:
x−24 = 0 or x+ 2 = 0
So the solutions are:
x= 24 or x=−2
Therefore, the solutions to the proportion 3x−1
x+7 =4x+5
2x−3are x= 24 and
x=−2.
Question 13
Question
Solve the proportion: 2x+3
x−1=x+4
x.
Solution
Step 1: Cross multiply to eliminate the fractions.
(x−1)(x+ 4) = (2x+ 3)(x)
x2+ 3x−4=2x2+ 3x
Step 2: Rearrange the equation to set it equal to zero.
0 = 2x2+ 3x−x2−3x−4
0 = x2−4
Step 3: Factor the quadratic equation.
0=(x+ 2)(x−2)
9
Step 4: Set each factor equal to zero and solve for x.
x+ 2 = 0 x−2=0
x=−2x= 2
Step 5: Check the solutions in the original proportion. Checking x=−2:
2(−2) + 3
−2−1=−2+4
−2
−1
−3=2
−2
1
3=−1
Step 6: Final answer. The solution to the proportion is x= 2.
Question 14
Question
In a certain bag of marbles, the ratio of blue marbles to red marbles is 3:5. If
there are a total of 64 marbles in the bag, how many red marbles are there?
Solution
Step 1: Let’s denote the number of blue marbles as 3xand the number of red
marbles as 5x, where xis a positive integer.
Step 2: We know that the total number of marbles is 64. Therefore, we have
the equation:
3x+ 5x= 64
Step 3: Simplifying the equation, we get:
8x= 64
Step 4: Dividing both sides of the equation by 8, we find:
x= 8
Step 5: Now, we can determine the number of red marbles by substituting
x= 8 back into our expression for the number of red marbles:
5x= 5(8) = 40
Step 6: Thus, there are 40 red marbles in the bag.
10
Question 15
Question
If xis directly proportional to yand inversely proportional to z, and x= 24
when y= 6 and z= 4, find the value of xwhen y= 8 and z= 6.
Solution
Let’s first express the proportionalities given:
xis directly proportional to y:x=ky for some constant k.
xis inversely proportional to z:x=k
zfor the same constant k.
Step 1: Find the value of kusing the first set of values given.
24 = k×6⇒k= 4
Step 2: Use the value of kto find the value of xwhen y= 8 and z= 6.
x= 4 ×8 = 32
Therefore, when y= 8 and z= 6, the value of xis 32.
Question 16
Question
Solve the following proportion for x:
3
x+ 2 =5
4x−1
Solution
To solve the proportion 3
x+2 =5
4x−1for x, we can cross-multiply to get rid of
the fractions.
Step 1: Cross-multiply to get:
3(4x−1) = 5(x+ 2)
Step 2: Expand both sides of the equation:
12x−3=5x+ 10
Step 3: Simplify the equation:
12x−5x= 10 + 3
11
7x= 13
Step 4: Divide both sides by 7 to solve for x:
x=13
7
Therefore, the solution to the proportion is x=13
7.
Question 17
Question
Solve the proportion: 4x
6=x+ 3
9
Solution
To solve the proportion, we will cross multiply and then solve for x.
Step 1: Cross multiply to get:
4x·9=6·(x+ 3)
Simplify both sides:
36x= 6x+ 18
Step 2: Move all terms involving xto one side of the equation:
36x−6x= 18
Simplify the left side of the equation:
30x= 18
Step 3: Solve for xby dividing both sides by 30:
x=18
30
Simplify the fraction:
x=3
5
Therefore, the solution to the proportion is x=3
5.
Question 18
Question
Solve the following proportion for x:
1
2=3
x+ 2
12
Solution
To solve the proportion given, we can cross multiply to eliminate the fractions
and then solve for x.
Step 1: Cross multiply the terms.
1(x+ 2) = 2 ·3
Step 2: Simplify both sides of the equation.
x+ 2 = 6
Step 3: Subtract 2 from both sides to solve for x.
x= 6 −2
Step 4: Calculate the value of x.
x= 4
Therefore, the value of xthat satisfies the proportion is 4.
Question 19
Question
Solve for xin the proportion 4
2x−1=x+ 3
5.
Solution
Step 1: Cross multiply to eliminate the fractions.
4·5 = (2x−1)(x+ 3)
20 = 2x2+ 6x−x−3
20 = 2x2+ 5x−3
Step 2: Rearrange the equation into standard quadratic form.
2x2+ 5x−3 = 20
2x2+ 5x−23 = 0
Step 3: Solve the quadratic equation using the quadratic formula: x=
−b±√b2−4ac
2a, where a= 2, b= 5, and c=−23.
x=−5±p52−4(2)(−23)
2(2)
x=−5±√25 + 184
4
x=−5±√209
4
13
Thus, the solutions for xare x=−5 + √209
4and x=−5−√209
4.
Question 20
Question
Solve the following proportion for x:3
x=5
12 .
Solution
Step 1: Cross multiply to solve for x.
3
x=5
12
3×12 = 5 ×x
36 = 5x
Step 2: Divide by 5 to solve for x.
x=36
5
x= 7.2
Therefore, the value of xis 7.2.
Question 21
Question
If 3x
2y=4y
5z, find x
zin terms of y.
Solution
Step 1: Multiply both sides of the equation by 2yz to clear the fractions.
3x
2y·2yz =4y
5z·2yz
⇒3xz = 8y
Step 2: Solve for x
zby isolating x
z.
x
z=8y
3
x
z=8y
3
14
Question 22
Question
Solve the proportion: 3x+4
5=8
x+2 .
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x+ 4)(x+ 2) = 5 ·8
3x2+ 10x+ 8 = 40
Step 2: Rearrange the equation and simplify.
3x2+ 10x+ 8 = 40
3x2+ 10x−32 = 0
Step 3: Factor the quadratic equation.
3x2+ 16x−6x−32 = 0
3x(x+ 5) −2(x+ 5) = 0
(3x−2)(x+ 5) = 0
Step 4: Solve for x. If 3x−2 = 0, then 3x= 2, x=2
3. If x+ 5 = 0, then
x=−5.
Therefore, the solutions to the proportion are x=2
3and x=−5.
Question 23
Question
Solve the following proportion for x:
7
10 =x
15
Solution
To solve the proportion 7
10 =x
15 for x, we can use the property that in a
proportion, the cross products are equal.
Step 1: Cross multiply to get:
7×15 = 10x
Step 2: Simplify the equation:
105 = 10x
15
Step 3: Divide by 10 to solve for x:
x=105
10 = 10.5
Therefore, the solution to the proportion is x= 10.5.
Question 24
Question
Solve the proportion: 3 + x
2=x+ 5
3.
Solution
Step 1: Cross-multiply to eliminate the fractions.
3(3 + x) = 2(x+ 5)
9+3x= 2x+ 10
Step 2: Move all terms involving xto one side of the equation.
3x−2x= 10 −9
x= 1
Step 3: Check the solution by substituting x= 1 back into the original
proportion. 3+1
2=1+5
3
4
2=6
3
2=2
Step 4: Therefore, the solution to the proportion is x= 1 .
Question 25
Question
If a:b= 5 : 8 and b:c= 2 : 3, find the ratio a:b:c.
Solution
Let us first find the value of bby setting up equations based on the given ratios.
16
Step 1: Find the value of b.
a
b=5
8
a=5
8b
b
c=2
3
c=3
2b
Step 2: Find the value of busing the two equations derived above.
a=5
8b(Substitute the value of a)
=5
83
2b
=15
16b
Therefore, b=16
15 a.
Step 3: Compile the ratio a:b:c.
a:b:c=a:16
15a:3
2·16
15a
= 15a: 16a: 32a
= 15 : 16 : 32
Question 26
Question
A recipe calls for 4 cups of flour to make 36 cookies. If you only have 1 cup of
flour, how many cookies can you make using the same ratio?
Solution
Let xrepresent the number of cookies that can be made using 1 cup of flour.
Step 1: Set up a proportion using the given information. By setting up a
proportion with the amount of flour and the number of cookies, we get:
4 cups of flour
36 cookies =1 cup of flour
xcookies
Step 2: Solve the proportion. Cross-multiplying, we get:
4·x= 36 ·1
17
4x= 36
x=36
4
x= 9
Step 3: Answer the question. Therefore, you can make 9 cookies using 1
cup of flour with the same ratio as in the original recipe.
Question 27
Question
Solve the following proportion for x:
3
4=2
x.
Solution
Step 1: Cross multiply to solve the proportion.
3·x= 4 ·2
3x= 8
Step 2: Divide both sides by 3 to solve for x.
3x
3=8
3
x=8
3
Therefore, the solution to the proportion 3
4=2
xis x=8
3.
Question 28
Question
If x:y= 3 : 7 and y:z= 5 : 4, find the ratio x:z.
Solution
Step 1: Let’s write the given ratios in fraction form:
x
y=3
7and y
z=5
4
Step 2: Since we want to find the ratio x:z, we need to find a relationship
between xand zin terms of y.
18
Step 3: From the given ratios, we can see that there is a common term y.
To eliminate y, we can equate the two expressions for y:
x
y=3
7=y
z=5
4
Step 4: Solving for yin terms of xand zfrom these equations, we get:
y=7x
3and y=5z
4
Step 5: Equating these two expressions for ygives us an equation involving
xand z:7x
3=5z
4
Step 6: Solving for x:zin terms of x, we find:
x
z=3
7×4
5=12
35
Step 7: Therefore, the ratio x:zis 12 : 35.
Question 29
Question
Solve the proportion: 3x−1
x+2 =5
4.
Solution
To solve the given proportion, we can cross multiply to get an equation in x.
Step 1: Cross multiply to get (3x−1) ·4=5·(x+ 2).
(3x−1) ·4 = 5 ·(x+ 2)
12x−4=5x+ 10
Step 2: Simplify the equation by combining like terms.
12x−4=5x+ 10
12x−5x= 10 + 4
7x= 14
Step 3: Solve for xby dividing both sides by 7.
7x= 14
7x
7=14
7
19
x= 2
Step 4: Check the solution by substituting x= 2 back into the original
proportion.
3(2) −1
2+2 =5
4
6−1
4=5
4
5
4=5
4
Since both sides of the proportion are equal with x= 2, our solution is
correct. Thus, x= 2 satisfies the given proportion.
Question 30
Question
Solve the following proportion for x:4x−1
2x+ 3 =5
6.
Solution
Step 1: Cross multiply to eliminate the fractions:
6(4x−1) = 5(2x+ 3)
24x−6 = 10x+ 15
Step 2: Simplify the equation by combining like terms:
24x−10x= 15 + 6
14x= 21
Step 3: Solve for xby dividing both sides by 14:
x=21
14
Step 4: Simplify the fraction to get the final answer:
x=3
2
Therefore, x=3
2.
20
Question 31
Question
A recipe for a certain dish requires 2 pounds of ingredients to produce 3 servings.
If you want to make 5 servings of the dish, how many pounds of ingredients will
you need?
Solution
Step 1: Calculate the ratio of pounds of ingredients to servings in the recipe.
Let the pounds of ingredients required be xfor 3 servings. So, the ratio of
pounds of ingredients to servings is 2/3 = x/3.
Step 2: Calculate the amount of ingredients required for 5 servings. Now we
need to find how many pounds of ingredients are needed for 5 servings. Using
the ratio from Step 1: 2/3 = x/3 Cross multiplying, we get: 3x= 2 ×3 3x= 6
x= 2 pounds
Therefore, to make 5 servings, you will need 3 pounds of ingredients.
Question 32
Question
A group of university students rented a van for a road trip. They agreed to split
the total cost equally. If 6 students bailed out at the last minute, each of the re-
maining students would need to pay 30more.Howmuchwasthetotalcostof rentingthevan?
Solution
Let xbe the total cost of renting the van and nbe the number of students who
initially agreed to split the cost. After 6 students bailed out, the number of
students left is n−6.
Step 1: Set up the equation based on the information provided:
x
n=x
n−6+ 30
Step 2: Clear the fractions by multiplying through by n(n−6):
x(n−6) = x·n+ 30n(n−6)
Step 3: Expand both sides of the equation:
xn −6x=xn + 30n2−180n
Step 4: Simplify the equation:
30n2−180n= 6x
21
5n2−30n=x
Step 5: Since the total cost is divided equally among the students, we have:
x=n·cost per student
Step 6: Substitute x= 5n2−30ninto x=n·cost per student:
5n2−30n=n·cost per student
Step 7: Solve for the cost per student:
cost per student = 5n2−30n
n= 5n−30
Step 8: From the given information, we know that after 6 students bailed out,
the remaining students had to pay 30more.T hisgivesustheequation : 5n−30 =
5n−30 + 30
Step 9: Solve for n:
5n−30 = 5n
−30 = 0
Step 10: This equation has no solution, which means our initial assumption
was incorrect. This can happen if the total cost of renting the van was not
properly divided among the students.
Question 33
Question
In a bakery, the ratio of the number of chocolate chip cookies to the number of
oatmeal raisin cookies is 4:7. If there are 72 oatmeal raisin cookies, how many
chocolate chip cookies are there?
Solution
Step 1: Determine the multiplier from the given ratio. Let the number of
chocolate chip cookies be represented by 4x, where x is the multiplier. Since
the ratio of chocolate chip cookies to oatmeal raisin cookies is 4:7, the number
of oatmeal raisin cookies is 7x.
Step 2: Use the information given in the problem to find the value of x. Given
that there are 72 oatmeal raisin cookies: 7x = 72 Solving for x: x=72
7= 102
7
Step 3: Calculate the number of chocolate chip cookies. Substitute the value
of x back into the expression for chocolate chip cookies: Number of chocolate
chip cookies = 4x Number of chocolate chip cookies = 4 ×102
7= 416
7
Therefore, there are 41 chocolate chip cookies in the bakery.
22
Question 34
Question
A bakery sells muffins and cookies in the ratio 3:4. If the bakery sells 210
muffins, how many cookies does it sell?
Solution
Let xbe the number of cookies sold by the bakery.
Step 1: Set up a proportion based on the ratio of muffins to cookies:
3
4=210
x
Step 2: Cross multiply to solve for x:
3·x= 4 ·210
Step 3: Simplify the equation:
3x= 840
Step 4: Divide both sides by 3 to solve for x:
x=840
3= 280
The bakery sells 280 cookies.
Question 35
Question
A group of 12 university students decided to share equally the cost of a trip.
Three students dropped out, and the remaining students had to increase their
contribution by $5 each to cover the cost. How much did the trip cost in total?
Solution
Step 1: Let’s denote the total cost of the trip as Tand each student’s initial
contribution as x. Since the cost is shared equally among 12 students:
12x=T
Step 2: After three students dropped out, there were only 9 students left.
Each student had to contribute an extra 5, so the total contribution per student
became x+ 5. The total cost was still T:
9(x+ 5) = T
23
Solution
Let’s begin solving the problem step by step:
Step 1: Write down the given ratio. Given that the ratio of xto yis 3 : 4,
we can express this as x
y=3
4.
Step 2: Set up the equation using the sum of 2xand 3y. Since the sum of
2xand 3yis 70, we can write this as: 2x+ 3y= 70.
Step 3: Use the ratio to express one variable in terms of the other. From
the ratio x
y=3
4, we can express xin terms of yas x=3
4y.
Step 4: Substitute the expression for xinto the equation from Step 2. Sub-
stituting x=3
4yinto 2x+ 3y= 70, we get: 2(3
4y)+3y= 70.
Step 5: Solve for y. Solving the equation gives: 6
4y+ 3y= 70
6y
4+ 3y= 70
6y+12y
4= 70
18y
4= 70
18y= 280
y=280
18
y= 15.56 (rounded to two decimal places).
Step 6: Find the value of xusing the ratio. Substitute y= 15.56 into x=3
4y
to find x:x=3
4×15.56
x= 11.67 (rounded to two decimal places).
Hence, the values of xand yare x= 11.67 and y= 15.56, respectively.
Question 3
Question
Given three numbers a,b, and csuch that a:b= 2 : 3 and b:c= 4 : 5, find
the ratio of ato c.
Solution
Let’s first express a,b, and cin terms of a common variable using the given
ratios.
Step 1: Express aand bin terms of k: Since a:b= 2 : 3, we can express a
and bas: a= 2kand b= 3k.
Step 2: Express band cin terms of m: Since b:c= 4 : 5, we can rewrite b
and cas: b= 4mand c= 5m.
Step 3: Find the ratio of ato c: Substitute the expressions for aand cin
terms of kand mrespectively: a
c=2k
5m.
Hence, the ratio of ato cis 2 : 5 .
2
Question 4
Question
Solve for xin the following proportion: 3x+2
x−4=5
2.
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x+ 2) ·2 = 5 ·(x−4)
6x+ 4 = 5x−20
Step 2: Subtract 5xfrom both sides.
6x+ 4 −5x= 5x−20 −5x
x+ 4 = −20
Step 3: Subtract 4 from both sides.
x+ 4 −4 = −20 −4
x=−24
Therefore, the solution is x=−24.
Question 5
Question
A construction company is building a bridge that is 200 meters long. The
company has calculated that it will take 10 days to complete the bridge if 20
workers are employed. If the company wants to finish the bridge in 6 days, how
many additional workers should they hire?
Solution
Step 1: Calculate the current rate at which the workers are completing the
bridge. Let Rbe the rate at which one worker completes the bridge in 10 days.
The total work to be completed is building a 200-meter bridge. Therefore, the
current rate of completion is given by:
Total work = R×20 ×10 = 200
Step 2: Solve for R.
R=200
200 = 1 meter/day
3
Step 3: Calculate the new rate needed to finish the bridge in 6 days. Let x
be the number of additional workers needed to complete the bridge in 6 days.
The new total work to be completed is still building a 200-meter bridge. The
new rate of completion when 20 + xworkers are employed is given by:
Total work = R×(20 + x)×6 = 200
Step 4: Solve for x.
1×(20 + x)×6 = 200
120 + 6x= 200
6x= 80
x=80
6= 13.3
Therefore, the construction company should hire 14 additional workers to
finish the bridge in 6 days.
Question 6
Question
Solve the following proportion: 3x+4
6x−8=5
7.
Solution
To solve the proportion 3x+4
6x−8=5
7, we need to cross multiply and solve for x.
Step 1: Cross multiply the fractions to obtain (3x+ 4) ·7 = 5 ·(6x−8).
21x+ 28 = 30x−40
Step 2: Rearrange the equation by moving all terms involving xto one side
of the equation.
28 + 40 = 30x−21x
68 = 9x
Step 3: Solve for xby dividing both sides by 9.
x=68
9
Therefore, the solution to the proportion 3x+4
6x−8=5
7is x=68
9.
Question 7
Question
If x, y, and zare positive real numbers such that x:y= 2 : 3 and y:z= 3 : 4,
find the ratio x:z.
4
Solution
We are given that x:y= 2 : 3 and y:z= 3 : 4. We need to find x:z.
Step 1: Write the ratios in fraction form. Given x:y= 2 : 3, we can write
this as x
y=2
3. Similarly, y:z= 3 : 4 can be written as y
z=3
4.
Step 2: Set up an equation using the fractions. Since x
y=2
3and y
z=3
4, we
can write x
y·y
z=2
3·3
4.
Step 3: Simplify the equation. Multiplying the fractions gives us x
z=1
2.
Step 4: Write the ratio x:z. Therefore, the ratio x:zis 1 : 2 .
Question 8
Question
Solve for xin the proportion: 3x
2=4
5x.
Solution
Step 1: Cross multiply to eliminate the fractions.
3x
2=4
5x
3x·5x= 2 ·4
Step 2: Simplify the equation.
15x2= 8
Step 3: Divide by 15 to isolate x2.
x2=8
15
Step 4: Take the square root of both sides to solve for x.
x=±r8
15
Step 5: Simplify the square root.
x=±2√15
5
Therefore, the solutions for xare x=2√15
5and x=−2√15
5.
Question 9
Question
If xis directly proportional to y2and inversely proportional to z, and x= 6
when y= 3 and z= 4, find xwhen y= 5 and z= 6.
5
Solution
Step 1: Write the equation of proportionality using the given information. Since
xis directly proportional to y2and inversely proportional to z, we have:
x=ky2·1
z
where kis a constant of proportionality.
Step 2: Use the initial conditions x= 6, y= 3, and z= 4 to find k.
Substitute these values into the equation to get:
6 = k·32·1
4
6=9k·1
4
6 = 9
4k
k=24
9=8
3
Step 3: Substitute the value of kinto the equation of proportionality. We
now have:
x=8
3y2·1
z
Step 4: Use the new values of y= 5 and z= 6 to find x. Substitute these
values into the equation:
x=8
3·52·1
6
x=8
3·25 ·1
6
x=200
3·1
6
x=200
18
x=100
9= 11.1
Therefore, when y= 5 and z= 6, x≈11.1.
Question 10
Question
If 8 men can complete a job in 12 days, and 6 women can complete the same
job in 18 days, how many days will it take for 12 men and 9 women working
together to complete the job?
6
Solution
Step 1: Calculate the man-day ratio for completing the job. Let Mbe the
number of men required to finish the job in 1 day. Given that 8 men complete
the job in 12 days,
M=8 men ×12 days
1= 96 man-days
Step 2: Calculate the woman-day ratio for completing the job. Let Wbe
the number of women required to finish the job in 1 day. Given that 6 women
complete the job in 18 days,
W=6 women ×18 days
1= 108 woman-days
Step 3: Calculate the combined man-day and woman-day ratio for complet-
ing the job together. Let Dbe the number of days required for 12 men and 9
women to complete the job together. Using the man-day and woman-day ratios:
12M+ 9W=D
12(96) + 9(108) = D
D= 1152 + 972 = 2124 man-woman days
Step 4: Determine the number of days needed to complete the job when
working together. Since 12 men and 9 women are working together,
D=12 men + 9 women
1 day ×tdays
2124 = 12 ×96 + 9 ×108 ×t
2124 = 1152 + 972t
972t= 972
t= 1 day
Therefore, it will take 12 men and 9 women working together 1 day to
complete the job.
Question 11
Question
Solve the following proportion for x:
3
x+ 1 =x
4
7
Solution
To solve the given proportion, we will cross-multiply and then solve for x.
Step 1: Cross-multiply to get rid of the fractions:
3·4 = x·(x+ 1)
Step 2: Simplify both sides of the equation:
12 = x2+x
Step 3: Rearrange the equation into a quadratic form:
x2+x−12 = 0
Step 4: Factor the quadratic equation:
(x+ 4)(x−3) = 0
Step 5: Set each factor to zero to solve for x:
For x+ 4 = 0:
x=−4
For x−3 = 0:
x= 3
Step 6: Check for extraneous solutions:
Since both x=−4 and x= 3 satisfy the original proportion, there are no
extraneous solutions.
Therefore, the solutions to the given proportion are x=−4 and x= 3.
Question 12
Question
Solve the following proportion for x:
3x−1
x+ 7 =4x+ 5
2x−3
Solution
To solve the proportion 3x−1
x+7 =4x+5
2x−3, we can cross multiply.
Step 1: Cross multiply to get rid of the fractions:
(3x−1)(2x−3) = (x+ 7)(4x+ 5)
Expanding both sides gives:
6x2−9x−2x+ 3 = 4x2+ 5x+ 28x+ 35
8
Simplify to get:
6x2−11x+ 3 = 4x2+ 33x+ 35
Step 2: Rearrange the equation to solve for x: Subtract 4x2, 33x, and 35
from both sides:
2x2−44x−32 = 0
Step 3: Solve the quadratic equation: To factor the quadratic equation
2x2−44x−32 = 0, we simplify by dividing everything by 2 to get:
x2−22x−16 = 0
The equation can be factored as:
(x−24)(x+ 2) = 0
Setting each factor to zero gives:
x−24 = 0 or x+ 2 = 0
So the solutions are:
x= 24 or x=−2
Therefore, the solutions to the proportion 3x−1
x+7 =4x+5
2x−3are x= 24 and
x=−2.
Question 13
Question
Solve the proportion: 2x+3
x−1=x+4
x.
Solution
Step 1: Cross multiply to eliminate the fractions.
(x−1)(x+ 4) = (2x+ 3)(x)
x2+ 3x−4=2x2+ 3x
Step 2: Rearrange the equation to set it equal to zero.
0 = 2x2+ 3x−x2−3x−4
0 = x2−4
Step 3: Factor the quadratic equation.
0=(x+ 2)(x−2)
9
Step 4: Set each factor equal to zero and solve for x.
x+ 2 = 0 x−2=0
x=−2x= 2
Step 5: Check the solutions in the original proportion. Checking x=−2:
2(−2) + 3
−2−1=−2+4
−2
−1
−3=2
−2
1
3=−1
Step 6: Final answer. The solution to the proportion is x= 2.
Question 14
Question
In a certain bag of marbles, the ratio of blue marbles to red marbles is 3:5. If
there are a total of 64 marbles in the bag, how many red marbles are there?
Solution
Step 1: Let’s denote the number of blue marbles as 3xand the number of red
marbles as 5x, where xis a positive integer.
Step 2: We know that the total number of marbles is 64. Therefore, we have
the equation:
3x+ 5x= 64
Step 3: Simplifying the equation, we get:
8x= 64
Step 4: Dividing both sides of the equation by 8, we find:
x= 8
Step 5: Now, we can determine the number of red marbles by substituting
x= 8 back into our expression for the number of red marbles:
5x= 5(8) = 40
Step 6: Thus, there are 40 red marbles in the bag.
10
Question 15
Question
If xis directly proportional to yand inversely proportional to z, and x= 24
when y= 6 and z= 4, find the value of xwhen y= 8 and z= 6.
Solution
Let’s first express the proportionalities given:
xis directly proportional to y:x=ky for some constant k.
xis inversely proportional to z:x=k
zfor the same constant k.
Step 1: Find the value of kusing the first set of values given.
24 = k×6⇒k= 4
Step 2: Use the value of kto find the value of xwhen y= 8 and z= 6.
x= 4 ×8 = 32
Therefore, when y= 8 and z= 6, the value of xis 32.
Question 16
Question
Solve the following proportion for x:
3
x+ 2 =5
4x−1
Solution
To solve the proportion 3
x+2 =5
4x−1for x, we can cross-multiply to get rid of
the fractions.
Step 1: Cross-multiply to get:
3(4x−1) = 5(x+ 2)
Step 2: Expand both sides of the equation:
12x−3=5x+ 10
Step 3: Simplify the equation:
12x−5x= 10 + 3
11
7x= 13
Step 4: Divide both sides by 7 to solve for x:
x=13
7
Therefore, the solution to the proportion is x=13
7.
Question 17
Question
Solve the proportion: 4x
6=x+ 3
9
Solution
To solve the proportion, we will cross multiply and then solve for x.
Step 1: Cross multiply to get:
4x·9=6·(x+ 3)
Simplify both sides:
36x= 6x+ 18
Step 2: Move all terms involving xto one side of the equation:
36x−6x= 18
Simplify the left side of the equation:
30x= 18
Step 3: Solve for xby dividing both sides by 30:
x=18
30
Simplify the fraction:
x=3
5
Therefore, the solution to the proportion is x=3
5.
Question 18
Question
Solve the following proportion for x:
1
2=3
x+ 2
12
Solution
To solve the proportion given, we can cross multiply to eliminate the fractions
and then solve for x.
Step 1: Cross multiply the terms.
1(x+ 2) = 2 ·3
Step 2: Simplify both sides of the equation.
x+ 2 = 6
Step 3: Subtract 2 from both sides to solve for x.
x= 6 −2
Step 4: Calculate the value of x.
x= 4
Therefore, the value of xthat satisfies the proportion is 4.
Question 19
Question
Solve for xin the proportion 4
2x−1=x+ 3
5.
Solution
Step 1: Cross multiply to eliminate the fractions.
4·5 = (2x−1)(x+ 3)
20 = 2x2+ 6x−x−3
20 = 2x2+ 5x−3
Step 2: Rearrange the equation into standard quadratic form.
2x2+ 5x−3 = 20
2x2+ 5x−23 = 0
Step 3: Solve the quadratic equation using the quadratic formula: x=
−b±√b2−4ac
2a, where a= 2, b= 5, and c=−23.
x=−5±p52−4(2)(−23)
2(2)
x=−5±√25 + 184
4
x=−5±√209
4
13
Thus, the solutions for xare x=−5 + √209
4and x=−5−√209
4.
Question 20
Question
Solve the following proportion for x:3
x=5
12 .
Solution
Step 1: Cross multiply to solve for x.
3
x=5
12
3×12 = 5 ×x
36 = 5x
Step 2: Divide by 5 to solve for x.
x=36
5
x= 7.2
Therefore, the value of xis 7.2.
Question 21
Question
If 3x
2y=4y
5z, find x
zin terms of y.
Solution
Step 1: Multiply both sides of the equation by 2yz to clear the fractions.
3x
2y·2yz =4y
5z·2yz
⇒3xz = 8y
Step 2: Solve for x
zby isolating x
z.
x
z=8y
3
x
z=8y
3
14
Question 22
Question
Solve the proportion: 3x+4
5=8
x+2 .
Solution
Step 1: Cross multiply to eliminate the fractions.
(3x+ 4)(x+ 2) = 5 ·8
3x2+ 10x+ 8 = 40
Step 2: Rearrange the equation and simplify.
3x2+ 10x+ 8 = 40
3x2+ 10x−32 = 0
Step 3: Factor the quadratic equation.
3x2+ 16x−6x−32 = 0
3x(x+ 5) −2(x+ 5) = 0
(3x−2)(x+ 5) = 0
Step 4: Solve for x. If 3x−2 = 0, then 3x= 2, x=2
3. If x+ 5 = 0, then
x=−5.
Therefore, the solutions to the proportion are x=2
3and x=−5.
Question 23
Question
Solve the following proportion for x:
7
10 =x
15
Solution
To solve the proportion 7
10 =x
15 for x, we can use the property that in a
proportion, the cross products are equal.
Step 1: Cross multiply to get:
7×15 = 10x
Step 2: Simplify the equation:
105 = 10x
15
Step 3: Divide by 10 to solve for x:
x=105
10 = 10.5
Therefore, the solution to the proportion is x= 10.5.
Question 24
Question
Solve the proportion: 3 + x
2=x+ 5
3.
Solution
Step 1: Cross-multiply to eliminate the fractions.
3(3 + x) = 2(x+ 5)
9+3x= 2x+ 10
Step 2: Move all terms involving xto one side of the equation.
3x−2x= 10 −9
x= 1
Step 3: Check the solution by substituting x= 1 back into the original
proportion. 3+1
2=1+5
3
4
2=6
3
2=2
Step 4: Therefore, the solution to the proportion is x= 1 .
Question 25
Question
If a:b= 5 : 8 and b:c= 2 : 3, find the ratio a:b:c.
Solution
Let us first find the value of bby setting up equations based on the given ratios.
16
Step 1: Find the value of b.
a
b=5
8
a=5
8b
b
c=2
3
c=3
2b
Step 2: Find the value of busing the two equations derived above.
a=5
8b(Substitute the value of a)
=5
83
2b
=15
16b
Therefore, b=16
15 a.
Step 3: Compile the ratio a:b:c.
a:b:c=a:16
15a:3
2·16
15a
= 15a: 16a: 32a
= 15 : 16 : 32
Question 26
Question
A recipe calls for 4 cups of flour to make 36 cookies. If you only have 1 cup of
flour, how many cookies can you make using the same ratio?
Solution
Let xrepresent the number of cookies that can be made using 1 cup of flour.
Step 1: Set up a proportion using the given information. By setting up a
proportion with the amount of flour and the number of cookies, we get:
4 cups of flour
36 cookies =1 cup of flour
xcookies
Step 2: Solve the proportion. Cross-multiplying, we get:
4·x= 36 ·1
17
4x= 36
x=36
4
x= 9
Step 3: Answer the question. Therefore, you can make 9 cookies using 1
cup of flour with the same ratio as in the original recipe.
Question 27
Question
Solve the following proportion for x:
3
4=2
x.
Solution
Step 1: Cross multiply to solve the proportion.
3·x= 4 ·2
3x= 8
Step 2: Divide both sides by 3 to solve for x.
3x
3=8
3
x=8
3
Therefore, the solution to the proportion 3
4=2
xis x=8
3.
Question 28
Question
If x:y= 3 : 7 and y:z= 5 : 4, find the ratio x:z.
Solution
Step 1: Let’s write the given ratios in fraction form:
x
y=3
7and y
z=5
4
Step 2: Since we want to find the ratio x:z, we need to find a relationship
between xand zin terms of y.
18
Step 3: From the given ratios, we can see that there is a common term y.
To eliminate y, we can equate the two expressions for y:
x
y=3
7=y
z=5
4
Step 4: Solving for yin terms of xand zfrom these equations, we get:
y=7x
3and y=5z
4
Step 5: Equating these two expressions for ygives us an equation involving
xand z:7x
3=5z
4
Step 6: Solving for x:zin terms of x, we find:
x
z=3
7×4
5=12
35
Step 7: Therefore, the ratio x:zis 12 : 35.
Question 29
Question
Solve the proportion: 3x−1
x+2 =5
4.
Solution
To solve the given proportion, we can cross multiply to get an equation in x.
Step 1: Cross multiply to get (3x−1) ·4=5·(x+ 2).
(3x−1) ·4 = 5 ·(x+ 2)
12x−4=5x+ 10
Step 2: Simplify the equation by combining like terms.
12x−4=5x+ 10
12x−5x= 10 + 4
7x= 14
Step 3: Solve for xby dividing both sides by 7.
7x= 14
7x
7=14
7
19
x= 2
Step 4: Check the solution by substituting x= 2 back into the original
proportion.
3(2) −1
2+2 =5
4
6−1
4=5
4
5
4=5
4
Since both sides of the proportion are equal with x= 2, our solution is
correct. Thus, x= 2 satisfies the given proportion.
Question 30
Question
Solve the following proportion for x:4x−1
2x+ 3 =5
6.
Solution
Step 1: Cross multiply to eliminate the fractions:
6(4x−1) = 5(2x+ 3)
24x−6 = 10x+ 15
Step 2: Simplify the equation by combining like terms:
24x−10x= 15 + 6
14x= 21
Step 3: Solve for xby dividing both sides by 14:
x=21
14
Step 4: Simplify the fraction to get the final answer:
x=3
2
Therefore, x=3
2.
20
Question 31
Question
A recipe for a certain dish requires 2 pounds of ingredients to produce 3 servings.
If you want to make 5 servings of the dish, how many pounds of ingredients will
you need?
Solution
Step 1: Calculate the ratio of pounds of ingredients to servings in the recipe.
Let the pounds of ingredients required be xfor 3 servings. So, the ratio of
pounds of ingredients to servings is 2/3 = x/3.
Step 2: Calculate the amount of ingredients required for 5 servings. Now we
need to find how many pounds of ingredients are needed for 5 servings. Using
the ratio from Step 1: 2/3 = x/3 Cross multiplying, we get: 3x= 2 ×3 3x= 6
x= 2 pounds
Therefore, to make 5 servings, you will need 3 pounds of ingredients.
Question 32
Question
A group of university students rented a van for a road trip. They agreed to split
the total cost equally. If 6 students bailed out at the last minute, each of the re-
maining students would need to pay 30more.Howmuchwasthetotalcostof rentingthevan?
Solution
Let xbe the total cost of renting the van and nbe the number of students who
initially agreed to split the cost. After 6 students bailed out, the number of
students left is n−6.
Step 1: Set up the equation based on the information provided:
x
n=x
n−6+ 30
Step 2: Clear the fractions by multiplying through by n(n−6):
x(n−6) = x·n+ 30n(n−6)
Step 3: Expand both sides of the equation:
xn −6x=xn + 30n2−180n
Step 4: Simplify the equation:
30n2−180n= 6x
21
5n2−30n=x
Step 5: Since the total cost is divided equally among the students, we have:
x=n·cost per student
Step 6: Substitute x= 5n2−30ninto x=n·cost per student:
5n2−30n=n·cost per student
Step 7: Solve for the cost per student:
cost per student = 5n2−30n
n= 5n−30
Step 8: From the given information, we know that after 6 students bailed out,
the remaining students had to pay 30more.T hisgivesustheequation : 5n−30 =
5n−30 + 30
Step 9: Solve for n:
5n−30 = 5n
−30 = 0
Step 10: This equation has no solution, which means our initial assumption
was incorrect. This can happen if the total cost of renting the van was not
properly divided among the students.
Question 33
Question
In a bakery, the ratio of the number of chocolate chip cookies to the number of
oatmeal raisin cookies is 4:7. If there are 72 oatmeal raisin cookies, how many
chocolate chip cookies are there?
Solution
Step 1: Determine the multiplier from the given ratio. Let the number of
chocolate chip cookies be represented by 4x, where x is the multiplier. Since
the ratio of chocolate chip cookies to oatmeal raisin cookies is 4:7, the number
of oatmeal raisin cookies is 7x.
Step 2: Use the information given in the problem to find the value of x. Given
that there are 72 oatmeal raisin cookies: 7x = 72 Solving for x: x=72
7= 102
7
Step 3: Calculate the number of chocolate chip cookies. Substitute the value
of x back into the expression for chocolate chip cookies: Number of chocolate
chip cookies = 4x Number of chocolate chip cookies = 4 ×102
7= 416
7
Therefore, there are 41 chocolate chip cookies in the bakery.
22
Question 34
Question
A bakery sells muffins and cookies in the ratio 3:4. If the bakery sells 210
muffins, how many cookies does it sell?
Solution
Let xbe the number of cookies sold by the bakery.
Step 1: Set up a proportion based on the ratio of muffins to cookies:
3
4=210
x
Step 2: Cross multiply to solve for x:
3·x= 4 ·210
Step 3: Simplify the equation:
3x= 840
Step 4: Divide both sides by 3 to solve for x:
x=840
3= 280
The bakery sells 280 cookies.
Question 35
Question
A group of 12 university students decided to share equally the cost of a trip.
Three students dropped out, and the remaining students had to increase their
contribution by $5 each to cover the cost. How much did the trip cost in total?
Solution
Step 1: Let’s denote the total cost of the trip as Tand each student’s initial
contribution as x. Since the cost is shared equally among 12 students:
12x=T
Step 2: After three students dropped out, there were only 9 students left.
Each student had to contribute an extra 5, so the total contribution per student
became x+ 5. The total cost was still T:
9(x+ 5) = T
23
Step 3: We now have a system of two equations:
(12x=T
9(x+ 5) = T
Let’s solve this system to find T.
Step 4: First, let’s solve the first equation for x:
12x=T
x=T
12
Step 5: Substitute x=T
12 into the second equation:
9T
12 + 5=T
Step 6: Simplify the equation:
9T
12 + 5=T
3T
4+ 45 = T
3T+ 180 = 4T
T= 180
Step 7: Therefore, the total cost of the trip was $180.
24
