MATH 114 - QUANTITATIVE
REASONING - Ratios and Proportions
Question Bank - Set 2
Liberty University
Question 1
Question
Solve for x:3x+1
5x−2=2
3.
Solution
Step 1: Cross multiply to get rid of the fractions:
3x+ 1
5x−2=2
3
(3x+ 1) ·3=2·(5x−2)
Step 2: Expand both sides of the equation:
9x+ 3 = 10x−4
Step 3: Rearrange the equation to isolate xon one side:
9x+ 3 = 10x−4
9x−10x=−4−3
−x=−7
Step 4: Finally, solve for x:
x=−7
−1
x= 7
Therefore, the solution to the equation 3x+1
5x−2=2
3is x= 7.
Question 2
Question
Solve the following proportion:
3x+ 2
4=5x−1
6
Solution
Step 1: Cross multiply to eliminate the fractions:
6(3x+ 2) = 4(5x−1)
18x+ 12 = 20x−4
Step 2: Rearrange the equation by moving all terms involving xto one side:
18x+ 12 = 20x−4
18x−20x=−4−12
−2x=−16
Step 3: Solve for x:
−2x=−16
x=−16
−2
x= 8
Therefore, the solution to the proportion is x= 8.
Question 3
Question
Solve the following proportion: 2x+3
5x−1=4x−1
3x+2 .
Solution
To solve the proportion, we will cross multiply and then solve for x.
Step 1: Cross multiply to get: (2x+ 3)(3x+ 2) = (5x−1)(4x−1).
Step 2: Expand both sides: 6x2+ 4x+ 9x+ 6 = 20x2−5x−4x+ 1
Step 3: Combine like terms on both sides: 6x2+ 13x+ 6 = 20x2−9x+ 1
Step 4: Subtract 6x2+ 13x+ 6 from both sides: 0 = 14x2−22x−5
Step 5: Factor the quadratic equation: (2x+ 1)(7x−5) = 0
Step 6: Set each factor equal to zero and solve for x: 2x+1 = 0 or 7x−5=0
Step 7: Solve for x: For 2x+ 1 = 0, we get 2x=−1
⇒x=−1
2.
2
For 7x−5 = 0, we get 7x= 5
⇒x=5
7.
Therefore, the solutions are x=−1
2and x=5
7.
Question 4
Question
If 5 pencils and 4 pens cost $24.00, and 3 pencils and 2 pens cost $12.80, what
is the cost of 1 pencil?
Solution
Step 1: Set up a system of equations based on the given information. Let xbe
the cost of 1 pencil and ybe the cost of 1 pen. Then we can write:
(5x+ 4y= 24.00
3x+ 2y= 12.80
Step 2: Solve the system of equations. We can solve this system using the
method of substitution or elimination. Let’s use substitution in this case. From
the second equation, we can solve for yto get:
y=12.80 −3x
2
Now substitute yinto the first equation:
5x+ 4 12.80 −3x
2= 24.00
Simplify the equation:
5x+ 4(6.40 −1.5x) = 24.00
5x+ 25.60 −6x= 24.00
−x=−1.6
x= 1.6
Step 3: Find the cost of 1 pencil. Therefore, the cost of 1 pencil is $1.60.
Question 5
Question
If 5 men can complete a construction project in 10 days, and 8 women can
complete the same project in 15 days, how many days will it take 3 men and 4
women working together to complete the project?
3
Solution
Let’s first find the rate at which each group works, in terms of fraction of the
project completed per day. Then we will use these rates to find the time it takes
for the combined group to finish the project.
Step 1: Find the rate at which one man works. Let xbe the fraction of the
project that one man can complete in one day. Given that 5 men can complete
the project in 10 days, we have:
5 men ×10 days = 1 project ⇒50x= 1 ⇒x=1
50
Therefore, one man can complete 1
50 of the project in one day.
Step 2: Find the rate at which one woman works. Let ybe the fraction of
the project that one woman can complete in one day. Given that 8 women can
complete the project in 15 days, we have:
8 women ×15 days = 1 project ⇒120y= 1 ⇒y=1
120
Therefore, one woman can complete 1
120 of the project in one day.
Step 3: Find the combined rate of 3 men and 4 women. The combined rate
is the sum of the rates of the men and women working together:
31
50+ 4 1
120=3
50 +1
30 =9+5
150 =14
150 =7
75
Therefore, the combined group can complete 7
75 of the project in one day.
Step 4: Find the time it takes for 3 men and 4 women to complete the
project. Let dbe the number of days it takes for 3 men and 4 women to
complete the project. Using the combined rate, we have:
d×7
75 = 1 ⇒d=75
7≈10.71 days
Therefore, it will take approximately 10.71 days for 3 men and 4 women
working together to complete the project.
Question 6
Question
If 5 men can complete a construction project in 10 days, how many days will it
take for 8 men to complete the same project?
Solution
Let’s denote the number of days it takes for 8 men to complete the project as
d. We can set up a proportion based on the information given:
4
5 men ×10 days
8 men ×ddays = 1
We can solve this proportion for d.
50
8d= 1
Step 1: Multiply both sides by 8dto isolate d.
50 = 8d
Step 2: Divide both sides by 8 to solve for d.
d=50
8= 6.25
Therefore, it will take 8 men approximately 6.25 days to complete the con-
struction project.
Question 7
Question
A recipe for a cake calls for 2 cups of flour and 3 eggs. If you want to make
4 cakes, how many cups of flour will you need if the number of eggs used is
doubled?
Solution
Let xrepresent the number of cups of flour needed for 4 cakes when the number
of eggs used is doubled.
Step 1: Determine the initial ratio of flour to eggs. For 1 cake: - Cups of
flour: 2 cups - Number of eggs: 3 eggs Thus, the ratio of cups of flour to eggs
is 2 : 3.
Step 2: Determine the new ratio of flour to eggs when the number of eggs
used is doubled. Since the number of eggs is doubled, the new ratio becomes: -
Cups of flour: xcups - Number of eggs: 3 ×2 = 6 eggs Hence, the new ratio of
cups of flour to eggs is x: 6.
Step 3: Set up a proportion to find the value of x:
2
3=x
6
Step 4: Solve for x:
2×6=3x
12 = 3x
x=12
3
5
x= 4
Therefore, you will need 4 cups of flour to make 4 cakes when the number
of eggs used is doubled.
Question 8
Question
In a bag of marbles, the ratio of red marbles to blue marbles is 3:4. If there are
28 red marbles, how many blue marbles are in the bag?
Solution
Let’s set up a proportion to find the number of blue marbles in the bag.
Step 1: Determine the ratio of red marbles to blue marbles The ratio of red
marbles to blue marbles is given as 3:4.
Step 2: Write the proportion using the given ratio and number of red mar-
bles Let xrepresent the number of blue marbles. We can set up the proportion:
3
4=28
x.
Step 3: Cross multiply to solve for xCross multiplying the proportion, we
get 3x= 4 ∗28.
Step 4: Solve for xSolving the equation, we find: 3x= 112
x=112
3
x= 371
3.
Thus, there are 37 blue marbles in the bag.
Question 9
Question
In a research study, the ratio of 1st-year students to 2nd-year students is 5:3. If
there are 240 1st-year students, how many 2nd-year students are there?
Solution
Step 1: Let xrepresent the number of 2nd-year students. Step 2: Use the given
ratio to set up the proportion: 240
x=5
3. Step 3: Cross multiply to solve for x:
240 ×3=5x. Step 4: Simplify the equation: 720 = 5x. Step 5: Solve for x:
x=720
5= 144. Step 6: There are 144 2nd-year students in the research study.
Question 10
Question
Solve for xin the proportion: 2
3=x+1
4x−3.
6
Solution
To solve for xin the proportion 2
3=x+1
4x−3, we will cross multiply.
Step 1: Cross multiply to get rid of the fractions.
2(4x−3) = 3(x+ 1)
8x−6 = 3x+ 3
Step 2: Rearrange the equation to isolate x.
8x−6=3x+ 3
8x−3x= 6 + 3
5x= 9
Step 3: Solve for xby dividing both sides by 5.
x=9
5
Therefore, the solution to the proportion is x=9
5.
Question 11
Question
Solve the following proportion for x:
3x+ 4
5=2x−1
3
Solution
To solve the proportion for x, we need to cross multiply and solve the resulting
equation.
Step 1: Cross multiply to get rid of the fractions:
(3x+ 4) ×3=5×(2x−1)
Step 2: Expand both sides of the equation:
9x+ 12 = 10x−5
Step 3: Rearrange the equation to isolate xon one side:
9x+ 12 = 10x−5
12 + 5 = 10x−9x
17 = x
Step 4: Therefore, the solution to the proportion is x= 17.
7
Question 12
Question
A recipe calls for 2 cups of sugar for every 3 cups of flour. If you want to make a
bigger batch of the recipe using 12 cups of flour, how many cups of sugar should
you use?
Solution
Step 1: Find the ratio of sugar to flour in the original recipe. Let xrepresent
the number of cups of sugar needed for 3 cups of flour. The ratio of sugar to
flour in the original recipe is 2
3, which means x
3=2
3. Solving for xgives x= 2.
Step 2: Determine the number of cups of sugar needed for 12 cups of flour.
Since for 3 cups of flour you need 2 cups of sugar, for 12 cups of flour you will
need 2
3×12 = 8 cups of sugar.
Therefore, you should use 8 cups of sugar when making a bigger batch of
the recipe with 12 cups of flour.
Question 13
Question
A recipe for a cake requires 2 cups of flour for every 3 eggs. If you want to make
a cake that uses 6 cups of flour, how many eggs should you use?
Solution
Step 1: Let’s set up a proportion using the given information. Let xrepresent
the number of eggs needed.
2 cups of flour
3 eggs =6 cups of flour
xeggs
Step 2: Cross multiply to solve for x.
2×x= 6 ×3
Step 3: Simplify the equation.
2x= 18
Step 4: Divide both sides by 2 to find the value of x.
x=18
2= 9
Therefore, you should use 9 eggs to make a cake that uses 6 cups of flour.
8
Question 14
Question
A recipe for banana bread calls for 2 cups of flour for every 3 bananas. If you
only have 7 bananas, how many cups of flour should you use to maintain the
same ratio?
Solution
Let the unknown number of cups of flour be represented by x. We can set up a
proportion based on the relationship between the cups of flour and the number
of bananas:
2
3=x
7
Step 1: Cross multiply to solve for x:
2×7=3x
14 = 3x
Step 2: Divide both sides by 3 to solve for x:
x=14
3= 42
3cups
Therefore, you should use 42
3cups of flour if you only have 7 bananas in
order to maintain the same ratio as the original recipe.
Question 15
Question
If a:b= 2 : 3 and b:c= 4 : 5, find a:b:c.
Solution
Step 1: We are given that a:b= 2 : 3 and b:c= 4 : 5.
Let’s represent these ratios using variables and constants:
a= 2x, b = 3x= 4y, c = 5y
where xand yare constants.
Step 2: To find the value of xand y, equate the expression for b:
3x= 4y
9
x=4
3y
Step 3: Since a= 2x, substitute the expression for xfound in Step 2:
a= 2 4
3y=8
3y
Step 4: Now we have the values of a,b, and cin terms of y. So the ratio
a:b:ccan be written as:
8
3y: 3x: 5y=8
3y: 3 4
3y: 5y=8:4:5
Step 5: Therefore, the ratio a:b:cis 8 : 4 : 5.
Question 16
Question
If 24 pounds of apples cost 36, howmuchdo48poundsofapplescost?
Solution
Let xrepresent the cost of 48 pounds of apples.
Step 1: Set up a ratio using the given information. The cost is directly
proportional to the weight of apples, so we can write the ratio:
24
36 =48
x
Step 2: Solve for xby cross multiplying:
24x= 36 ×48
24x= 1728
Step 3: Divide by 24 to solve for x:
x=1728
24
x= 72
Therefore, 48 pounds of apples cost 72.
Question 17
Question
Solve for xin the following proportion: 2x
3=x−1
2.
10
Solution
To solve this proportion, we will cross multiply and solve for x.
Step 1: Cross multiply the terms in the proportion.
2x·2=3·(x−1)
4x= 3x−3
Step 2: Subtract 3xfrom both sides of the equation.
4x−3x= 3x−3−3x
x=−3
Step 3: Check the solution by substituting x=−3 back into the original
proportion.
2(−3)
3=(−3) −1
2
−6
3=−4
2
−2 = −2
Therefore, the solution to the proportion is x=−3.
Question 18
Question
Solve for xin the following proportion:
3x+ 5
4x−1=7
9
Solution
Step 1: Cross multiply to get rid of the fractions.
9(3x+ 5) = 7(4x−1)
27x+ 45 = 28x−7
Step 2: Simplify the equation.
27x+ 45 = 28x−7
45 + 7 = 28x−27x
52 = x
Conclusion: The solution to the given proportion is x= 52.
11
Question 19
Question
Given that a:b= 3 : 2 and b:c= 4 : 5, find the ratio a:b:c.
Solution
We are given that a:b= 3 : 2 and b:c= 4 : 5. To find the ratio a:b:c, we
need to express a,b, and cin terms of the given ratios and then combine them.
Step 1: Find the value of bSince a:b= 3 : 2 and b:c= 4 : 5, we can
equate the second terms to find the value of b:
a
b=3
2and b
c=4
5
Cross multiplying gives us:
2a= 3band 4c= 5b
Solving these two equations simultaneously, we find:
b=2
3a=4
5c
Step 2: Express a,b, and cin terms of the common factor Let’s
express aand cin terms of b:
a=3
2b
c=5
4b
Step 3: Write the ratio a:b:cNow that we have expressions for a,b,
and cin terms of b, we can write the ratio a:b:cas:
a:b:c=3
2b:b:5
4b=3:3
2:5=6:3:10
Therefore, the ratio a:b:cis 6 : 3 : 10 .
Question 20
Question
Solve for x:3x
4=5
7.
12
Solution
Step 1: Cross multiply to eliminate the fractions.
3x
4=5
7
3x×7=4×5
Step 2: Simplify the expression.
21x= 20
Step 3: Solve for xby dividing both sides by 21.
x=20
21
Therefore, the solution is x=20
21 .
Question 21
Question
In a survey, it was found that 45
Solution
Let xbe the number of respondents who preferred brand A.
Step 1: Write the proportion using the given information:
x
630 =45
100
Step 2: Simplify the proportion by dividing both sides by 100:
x
630 = 0.45
Step 3: Solve for xby multiplying both sides by 630:
x= 0.45 ×630
Step 4: Calculate the number of respondents who preferred brand A:
x= 283.5
Step 5: Since we cannot have a fraction of a respondent, round up to the
nearest whole number:
x≈284
Therefore, 284 respondents preferred brand A out of the 630 total respon-
dents.
13
Question 22
Question
Solve the following proportion for x:
3
2=4
x+ 1.
Solution
Step 1: Cross multiply to eliminate the fractions:
3(x+ 1) = 2 ·4
3x+ 3 = 8
Step 2: Subtract 3 from both sides:
3x= 8 −3
3x= 5
Step 3: Divide by 3 on both sides to solve for x:
x=5
3
Thus, the solution to the proportion is x=5
3.
Question 23
Question
Solve the following proportion for x:3
4=x−1
2x+1 .
Solution
To solve the proportion 3
4=x−1
2x+1 for x, we can cross multiply to eliminate the
fractions.
Step 1: Cross multiply to get rid of the fractions.
3(2x+ 1) = 4(x−1)
Step 2: Expand both sides of the equation.
6x+ 3 = 4x−4
Step 3: Rearrange the equation by isolating the variable terms on one side.
6x−4x=−4−3
14
2x=−7
Step 4: Solve for xby dividing both sides by 2.
x=−7
2
Step 5: Therefore, the solution to the proportion 3
4=x−1
2x+1 is x=−7
2.
Question 24
Question
If 6 men can complete a construction project in 10 days, and 8 women can
complete the same project in 15 days, how many days will it take for 3 men and
4 women working together to complete the project?
Solution
Step 1: Let’s first find the work rates of men and women separately. Let mbe
the work rate of a man and wbe the work rate of a woman. From the given
information, we have: 6 men complete the project in 10 days =⇒6m=1
10 =⇒
m=1
60 . 8 women complete the project in 15 days =⇒8w=1
15 =⇒w=1
120 .
Step 2: Next, let’s find the work rate when 3 men and 4 women work to-
gether. The work rate of 3 men and 4 women working together is (3m+ 4w),
which is equal to 3×1
60 + 4 ×1
120 =1
20 .
Step 3: Finally, let’s find how many days it will take for 3 men and 4 women
to complete the project together. Let dbe the number of days required. By the
work-rate formula Work rate = 1
time , we have
1
20 =1
d=⇒d= 20 days
Therefore, it will take 3 men and 4 women working together 20 days to
complete the project.
Question 25
Question
Solve the following proportion for x:
3
x+ 1 =x+ 2
5
15
Solution
Step 1: Cross multiply to eliminate the fractions.
3·5=(x+ 1)(x+ 2)
Step 2: Simplify and expand the expression.
15 = x2+ 3x+ 2
Step 3: Rearrange the quadratic equation to set it equal to zero.
x2+ 3x+ 2 −15 = 0
Step 4: Combine like terms.
x2+ 3x−13 = 0
Step 5: Solve the quadratic equation using the quadratic formula:
x=−b±√b2−4ac
2a
Step 6: Substitute a= 1, b= 3, and c=−13.
x=−3±p32−4·1·(−13)
2·1
Step 7: Calculate the discriminant under the square root.
x=−3±√9 + 52
2
Step 8: Simplify the expression further.
x=−3±√61
2
Therefore, the solutions for xare x=−3+√61
2and x=−3−√61
2.
Question 26
Question
In a certain company, the ratio of managers to non-managers is 3:7. If there are
100 more non-managers than managers, how many managers are there in the
company?
16
Solution
Let’s denote the number of managers as 3xand the number of non-managers as
7x. We are given that 7x= 3x+ 100. Step 1: Set up the equation based on
the given information.
Since we are given that there are 100 more non-managers than managers, we
have the equation: 7x= 3x+ 100.
Step 2: Solve the equation for x.
Subtracting 3xfrom both sides, we get: 4x= 100.
Dividing by 4, we find: x= 25.
Step 3: Calculate the number of managers.
Now, we can find the number of managers: 3x= 3(25) = 75.
Step 4: Check the answer.
To confirm our solution, we can calculate the number of non-managers: 7x=
7(25) = 175, which is indeed 100 more than the number of managers.
Therefore, there are 75 managers in the company.
Question 27
Question
A group of students decided to share a certain number of pencils. If each student
receives 4 pencils more than they planned, the number of pencils each student
receives will be increased by 40
Solution
Let’s denote the original number of students as xand the original number of
pencils as y.
Step 1: Translate the problem into equations If each student receives
4 pencils more than they planned, the number of pencils each student receives
will be increased by 40
1.4y=x(y+ 4)
If they had originally planned for 12 students more, the number of students
would have been x+ 12. The number of pencils to be shared remains the same,
so we get:
x(y+ 4) = (x+ 12)y
Step 2: Solve the system of equations Now, we solve the system of
equations:
(1.4y=x(y+ 4)
x(y+ 4) = (x+ 12)y
First, we simplify the equations:
1.4y=xy + 4x(1)
17
xy + 4x=xy + 12y(2)
Subtracting equation (1) from equation (2) gives:
4x= 12y=⇒x= 3y
Step 3: Substitute x= 3yinto one of the original equations Substi-
tute x= 3yinto equation (1):
1.4y= 3y(y+ 4)
1.4y= 3y2+ 12y
3y2−1.6y= 0
y(3y−1.6) = 0
Since ycannot be zero, we have:
3y−1.6 = 0 =⇒y=1.6
3=8
15
Step 4: Find the value of xSubstitute y=8
15 back into x= 3y:
x= 3 ×8
15 =8
5= 1.6
Therefore, the initial number of students was 1.6, which is not possible. This
means there was a mistake in the problem setup.
Question 28
Question
Solve the following proportion for xif 2
x+5 =x−1
3.
Solution
Step 1: Cross multiply to eliminate the fractions.
2(3) = (x+ 5)(x−1)
6 = x2+ 4x−5
Step 2: Rearrange the equation into standard form.
x2+ 4x−11 = 0
Step 3: Use the quadratic formula to solve for x.
x=−b±√b2−4ac
2a
18
Step 4: Substitute a= 1, b= 4, and c=−11 into the formula.
x=−4±p42−4(1)(−11)
2(1)
Step 5: Simplify the expression under the square root.
x=−4±√16 + 44
2
x=−4±√60
2
x=−4±2√15
2
Step 6: Simplify the expression by factoring out a 2 from the numerator.
x=2(−2±√15)
2
Step 7: Cancel out the common factor of 2.
x=−2±√15
Therefore, the solutions for xare x=−2 + √15 and x=−2−√15.
Question 29
Question
Solve for x:3
x+ 2 =5
x−3.
Solution
Step 1: Cross multiply to get rid of the fractions.
3(x−3) = 5(x+ 2)
3x−9=5x+ 10
Step 2: Rearrange the equation by isolating xterms on one side.
3x−5x= 10 + 9
−2x= 19
Step 3: Solve for xby dividing both sides by −2.
x=19
−2
x=−19
2
Therefore, the solution to the equation is x=−19
2.
19
Question 30
Question
Solve the following proportion for x:
3
2x−1=5
4x+ 6
Solution
Let’s cross multiply to solve the proportion:
Step 1: Cross multiply to obtain:
3(4x+ 6) = 5(2x−1)
Step 2: Expand both sides of the equation:
12x+ 18 = 10x−5
Step 3: Rearrange the equation by moving all xterms to the left side:
12x−10x=−5−18
2x=−23
Step 4: Finally, divide both sides by 2 to solve for x:
x=−23
2
Therefore, the solution to the proportion is x=−23
2.
Question 31
Question
Solve the following proportion: 5x
6=2x+3
9.
Solution
Step 1: Cross multiply to eliminate the fractions.
5x
6=2x+ 3
9
9·5x= 6 ·(2x+ 3)
Step 2: Simplify both sides of the equation.
45x= 12x+ 18
20
Step 3: Move all terms involving xto one side of the equation.
45x−12x= 18
33x= 18
Step 4: Solve for xby dividing both sides by 33.
x=18
33 =6
11
Therefore, the solution to the proportion is x=6
11 .
Question 32
Question
If 40
Solution
Step 1: Let’s denote the two numbers in the ratio as 2xand 5x.
Step 2: We know that 40
40
100(2x) = 30
100(5x)
Step 3: Simplifying the equation, we get:
0.4(2x)=0.3(5x)
0.8x= 1.5x
Step 4: Solving for x, we find:
1.5x−0.8x= 0
0.7x= 0
x= 0
Step 5: Since x= 0, the numbers are 2(0) = 0 and 5(0) = 0.
Step 6: Therefore, the larger number is 0 .
Question 33
Question
Simplify the following expression: 2x2−6x
4x2−12 .
21
Solution
Step 1: Factor out a 2xfrom the numerator and a 4 from the denominator.
2x2−6x
4x2−12 =2x(x−3)
4(x2−3)
Step 2: Simplify by canceling out common factors.
2x(x−3)
4(x2−3) =2(x−3)
2(x2−3) =x−3
x2−3
Therefore, the simplified expression is x−3
x2−3.
Question 34
Question
If a certain quantity is divided into three parts that are in the ratio 3 : 4 : 5,
and the smallest part is 12, what is the quantity?
Solution
Let’s denote the quantity as Q. Since the parts are in the ratio 3 : 4 : 5, we can
express the parts as 3x, 4x, and 5x, where xis a constant of proportionality.
Step 1: Write the equation based on the information given. From the
problem, we know that the smallest part is 12. This means that 3x= 12. We
can solve for xto find the value of each part.
3x= 12
x= 4
Step 2: Find the value of the quantity. Now that we know the value of x,
we can find the three parts:
3x= 3(4) = 12
4x= 4(4) = 16
5x= 5(4) = 20
The total quantity, Q, is the sum of these three parts:
Q= 3x+ 4x+ 5x
Q= 12 + 16 + 20
Q= 48
Therefore, the quantity is 48.
22
Question 35
Question
A recipe for chocolate chip cookies requires 1 1/2 cups of flour for every 1 cup of
sugar. If you want to make a batch of cookies using 3 cups of flour, how many
cups of sugar should you use?
Solution
Let xbe the number of cups of sugar needed to make the batch of cookies using
3 cups of flour.
Step 1: Set up a proportion using the given ratio:
11
2
1=3
x
Step 2: Simplify the proportion:
3/2
1=3
x
3
2=3
x
Step 3: Cross multiply to solve for x:
3x= 2 ×3
3x= 6
Step 4: Divide by 3 to solve for x:
x=6
3
x= 2
Therefore, you should use 2 cups of sugar to make a batch of cookies using
3 cups of flour.
23
Question 2
Question
Solve the following proportion:
3x+ 2
4=5x−1
6
Solution
Step 1: Cross multiply to eliminate the fractions:
6(3x+ 2) = 4(5x−1)
18x+ 12 = 20x−4
Step 2: Rearrange the equation by moving all terms involving xto one side:
18x+ 12 = 20x−4
18x−20x=−4−12
−2x=−16
Step 3: Solve for x:
−2x=−16
x=−16
−2
x= 8
Therefore, the solution to the proportion is x= 8.
Question 3
Question
Solve the following proportion: 2x+3
5x−1=4x−1
3x+2 .
Solution
To solve the proportion, we will cross multiply and then solve for x.
Step 1: Cross multiply to get: (2x+ 3)(3x+ 2) = (5x−1)(4x−1).
Step 2: Expand both sides: 6x2+ 4x+ 9x+ 6 = 20x2−5x−4x+ 1
Step 3: Combine like terms on both sides: 6x2+ 13x+ 6 = 20x2−9x+ 1
Step 4: Subtract 6x2+ 13x+ 6 from both sides: 0 = 14x2−22x−5
Step 5: Factor the quadratic equation: (2x+ 1)(7x−5) = 0
Step 6: Set each factor equal to zero and solve for x: 2x+1 = 0 or 7x−5=0
Step 7: Solve for x: For 2x+ 1 = 0, we get 2x=−1
⇒x=−1
2.
2
For 7x−5 = 0, we get 7x= 5
⇒x=5
7.
Therefore, the solutions are x=−1
2and x=5
7.
Question 4
Question
If 5 pencils and 4 pens cost $24.00, and 3 pencils and 2 pens cost $12.80, what
is the cost of 1 pencil?
Solution
Step 1: Set up a system of equations based on the given information. Let xbe
the cost of 1 pencil and ybe the cost of 1 pen. Then we can write:
(5x+ 4y= 24.00
3x+ 2y= 12.80
Step 2: Solve the system of equations. We can solve this system using the
method of substitution or elimination. Let’s use substitution in this case. From
the second equation, we can solve for yto get:
y=12.80 −3x
2
Now substitute yinto the first equation:
5x+ 4 12.80 −3x
2= 24.00
Simplify the equation:
5x+ 4(6.40 −1.5x) = 24.00
5x+ 25.60 −6x= 24.00
−x=−1.6
x= 1.6
Step 3: Find the cost of 1 pencil. Therefore, the cost of 1 pencil is $1.60.
Question 5
Question
If 5 men can complete a construction project in 10 days, and 8 women can
complete the same project in 15 days, how many days will it take 3 men and 4
women working together to complete the project?
3
Solution
Let’s first find the rate at which each group works, in terms of fraction of the
project completed per day. Then we will use these rates to find the time it takes
for the combined group to finish the project.
Step 1: Find the rate at which one man works. Let xbe the fraction of the
project that one man can complete in one day. Given that 5 men can complete
the project in 10 days, we have:
5 men ×10 days = 1 project ⇒50x= 1 ⇒x=1
50
Therefore, one man can complete 1
50 of the project in one day.
Step 2: Find the rate at which one woman works. Let ybe the fraction of
the project that one woman can complete in one day. Given that 8 women can
complete the project in 15 days, we have:
8 women ×15 days = 1 project ⇒120y= 1 ⇒y=1
120
Therefore, one woman can complete 1
120 of the project in one day.
Step 3: Find the combined rate of 3 men and 4 women. The combined rate
is the sum of the rates of the men and women working together:
31
50+ 4 1
120=3
50 +1
30 =9+5
150 =14
150 =7
75
Therefore, the combined group can complete 7
75 of the project in one day.
Step 4: Find the time it takes for 3 men and 4 women to complete the
project. Let dbe the number of days it takes for 3 men and 4 women to
complete the project. Using the combined rate, we have:
d×7
75 = 1 ⇒d=75
7≈10.71 days
Therefore, it will take approximately 10.71 days for 3 men and 4 women
working together to complete the project.
Question 6
Question
If 5 men can complete a construction project in 10 days, how many days will it
take for 8 men to complete the same project?
Solution
Let’s denote the number of days it takes for 8 men to complete the project as
d. We can set up a proportion based on the information given:
4
5 men ×10 days
8 men ×ddays = 1
We can solve this proportion for d.
50
8d= 1
Step 1: Multiply both sides by 8dto isolate d.
50 = 8d
Step 2: Divide both sides by 8 to solve for d.
d=50
8= 6.25
Therefore, it will take 8 men approximately 6.25 days to complete the con-
struction project.
Question 7
Question
A recipe for a cake calls for 2 cups of flour and 3 eggs. If you want to make
4 cakes, how many cups of flour will you need if the number of eggs used is
doubled?
Solution
Let xrepresent the number of cups of flour needed for 4 cakes when the number
of eggs used is doubled.
Step 1: Determine the initial ratio of flour to eggs. For 1 cake: - Cups of
flour: 2 cups - Number of eggs: 3 eggs Thus, the ratio of cups of flour to eggs
is 2 : 3.
Step 2: Determine the new ratio of flour to eggs when the number of eggs
used is doubled. Since the number of eggs is doubled, the new ratio becomes: -
Cups of flour: xcups - Number of eggs: 3 ×2 = 6 eggs Hence, the new ratio of
cups of flour to eggs is x: 6.
Step 3: Set up a proportion to find the value of x:
2
3=x
6
Step 4: Solve for x:
2×6=3x
12 = 3x
x=12
3
5
x= 4
Therefore, you will need 4 cups of flour to make 4 cakes when the number
of eggs used is doubled.
Question 8
Question
In a bag of marbles, the ratio of red marbles to blue marbles is 3:4. If there are
28 red marbles, how many blue marbles are in the bag?
Solution
Let’s set up a proportion to find the number of blue marbles in the bag.
Step 1: Determine the ratio of red marbles to blue marbles The ratio of red
marbles to blue marbles is given as 3:4.
Step 2: Write the proportion using the given ratio and number of red mar-
bles Let xrepresent the number of blue marbles. We can set up the proportion:
3
4=28
x.
Step 3: Cross multiply to solve for xCross multiplying the proportion, we
get 3x= 4 ∗28.
Step 4: Solve for xSolving the equation, we find: 3x= 112
x=112
3
x= 371
3.
Thus, there are 37 blue marbles in the bag.
Question 9
Question
In a research study, the ratio of 1st-year students to 2nd-year students is 5:3. If
there are 240 1st-year students, how many 2nd-year students are there?
Solution
Step 1: Let xrepresent the number of 2nd-year students. Step 2: Use the given
ratio to set up the proportion: 240
x=5
3. Step 3: Cross multiply to solve for x:
240 ×3=5x. Step 4: Simplify the equation: 720 = 5x. Step 5: Solve for x:
x=720
5= 144. Step 6: There are 144 2nd-year students in the research study.
Question 10
Question
Solve for xin the proportion: 2
3=x+1
4x−3.
6
Solution
To solve for xin the proportion 2
3=x+1
4x−3, we will cross multiply.
Step 1: Cross multiply to get rid of the fractions.
2(4x−3) = 3(x+ 1)
8x−6 = 3x+ 3
Step 2: Rearrange the equation to isolate x.
8x−6=3x+ 3
8x−3x= 6 + 3
5x= 9
Step 3: Solve for xby dividing both sides by 5.
x=9
5
Therefore, the solution to the proportion is x=9
5.
Question 11
Question
Solve the following proportion for x:
3x+ 4
5=2x−1
3
Solution
To solve the proportion for x, we need to cross multiply and solve the resulting
equation.
Step 1: Cross multiply to get rid of the fractions:
(3x+ 4) ×3=5×(2x−1)
Step 2: Expand both sides of the equation:
9x+ 12 = 10x−5
Step 3: Rearrange the equation to isolate xon one side:
9x+ 12 = 10x−5
12 + 5 = 10x−9x
17 = x
Step 4: Therefore, the solution to the proportion is x= 17.
7
Question 12
Question
A recipe calls for 2 cups of sugar for every 3 cups of flour. If you want to make a
bigger batch of the recipe using 12 cups of flour, how many cups of sugar should
you use?
Solution
Step 1: Find the ratio of sugar to flour in the original recipe. Let xrepresent
the number of cups of sugar needed for 3 cups of flour. The ratio of sugar to
flour in the original recipe is 2
3, which means x
3=2
3. Solving for xgives x= 2.
Step 2: Determine the number of cups of sugar needed for 12 cups of flour.
Since for 3 cups of flour you need 2 cups of sugar, for 12 cups of flour you will
need 2
3×12 = 8 cups of sugar.
Therefore, you should use 8 cups of sugar when making a bigger batch of
the recipe with 12 cups of flour.
Question 13
Question
A recipe for a cake requires 2 cups of flour for every 3 eggs. If you want to make
a cake that uses 6 cups of flour, how many eggs should you use?
Solution
Step 1: Let’s set up a proportion using the given information. Let xrepresent
the number of eggs needed.
2 cups of flour
3 eggs =6 cups of flour
xeggs
Step 2: Cross multiply to solve for x.
2×x= 6 ×3
Step 3: Simplify the equation.
2x= 18
Step 4: Divide both sides by 2 to find the value of x.
x=18
2= 9
Therefore, you should use 9 eggs to make a cake that uses 6 cups of flour.
8
Question 14
Question
A recipe for banana bread calls for 2 cups of flour for every 3 bananas. If you
only have 7 bananas, how many cups of flour should you use to maintain the
same ratio?
Solution
Let the unknown number of cups of flour be represented by x. We can set up a
proportion based on the relationship between the cups of flour and the number
of bananas:
2
3=x
7
Step 1: Cross multiply to solve for x:
2×7=3x
14 = 3x
Step 2: Divide both sides by 3 to solve for x:
x=14
3= 42
3cups
Therefore, you should use 42
3cups of flour if you only have 7 bananas in
order to maintain the same ratio as the original recipe.
Question 15
Question
If a:b= 2 : 3 and b:c= 4 : 5, find a:b:c.
Solution
Step 1: We are given that a:b= 2 : 3 and b:c= 4 : 5.
Let’s represent these ratios using variables and constants:
a= 2x, b = 3x= 4y, c = 5y
where xand yare constants.
Step 2: To find the value of xand y, equate the expression for b:
3x= 4y
9
x=4
3y
Step 3: Since a= 2x, substitute the expression for xfound in Step 2:
a= 2 4
3y=8
3y
Step 4: Now we have the values of a,b, and cin terms of y. So the ratio
a:b:ccan be written as:
8
3y: 3x: 5y=8
3y: 3 4
3y: 5y=8:4:5
Step 5: Therefore, the ratio a:b:cis 8 : 4 : 5.
Question 16
Question
If 24 pounds of apples cost 36, howmuchdo48poundsofapplescost?
Solution
Let xrepresent the cost of 48 pounds of apples.
Step 1: Set up a ratio using the given information. The cost is directly
proportional to the weight of apples, so we can write the ratio:
24
36 =48
x
Step 2: Solve for xby cross multiplying:
24x= 36 ×48
24x= 1728
Step 3: Divide by 24 to solve for x:
x=1728
24
x= 72
Therefore, 48 pounds of apples cost 72.
Question 17
Question
Solve for xin the following proportion: 2x
3=x−1
2.
10
Solution
To solve this proportion, we will cross multiply and solve for x.
Step 1: Cross multiply the terms in the proportion.
2x·2=3·(x−1)
4x= 3x−3
Step 2: Subtract 3xfrom both sides of the equation.
4x−3x= 3x−3−3x
x=−3
Step 3: Check the solution by substituting x=−3 back into the original
proportion.
2(−3)
3=(−3) −1
2
−6
3=−4
2
−2 = −2
Therefore, the solution to the proportion is x=−3.
Question 18
Question
Solve for xin the following proportion:
3x+ 5
4x−1=7
9
Solution
Step 1: Cross multiply to get rid of the fractions.
9(3x+ 5) = 7(4x−1)
27x+ 45 = 28x−7
Step 2: Simplify the equation.
27x+ 45 = 28x−7
45 + 7 = 28x−27x
52 = x
Conclusion: The solution to the given proportion is x= 52.
11
Question 19
Question
Given that a:b= 3 : 2 and b:c= 4 : 5, find the ratio a:b:c.
Solution
We are given that a:b= 3 : 2 and b:c= 4 : 5. To find the ratio a:b:c, we
need to express a,b, and cin terms of the given ratios and then combine them.
Step 1: Find the value of bSince a:b= 3 : 2 and b:c= 4 : 5, we can
equate the second terms to find the value of b:
a
b=3
2and b
c=4
5
Cross multiplying gives us:
2a= 3band 4c= 5b
Solving these two equations simultaneously, we find:
b=2
3a=4
5c
Step 2: Express a,b, and cin terms of the common factor Let’s
express aand cin terms of b:
a=3
2b
c=5
4b
Step 3: Write the ratio a:b:cNow that we have expressions for a,b,
and cin terms of b, we can write the ratio a:b:cas:
a:b:c=3
2b:b:5
4b=3:3
2:5=6:3:10
Therefore, the ratio a:b:cis 6 : 3 : 10 .
Question 20
Question
Solve for x:3x
4=5
7.
12
Solution
Step 1: Cross multiply to eliminate the fractions.
3x
4=5
7
3x×7=4×5
Step 2: Simplify the expression.
21x= 20
Step 3: Solve for xby dividing both sides by 21.
x=20
21
Therefore, the solution is x=20
21 .
Question 21
Question
In a survey, it was found that 45
Solution
Let xbe the number of respondents who preferred brand A.
Step 1: Write the proportion using the given information:
x
630 =45
100
Step 2: Simplify the proportion by dividing both sides by 100:
x
630 = 0.45
Step 3: Solve for xby multiplying both sides by 630:
x= 0.45 ×630
Step 4: Calculate the number of respondents who preferred brand A:
x= 283.5
Step 5: Since we cannot have a fraction of a respondent, round up to the
nearest whole number:
x≈284
Therefore, 284 respondents preferred brand A out of the 630 total respon-
dents.
13
Question 22
Question
Solve the following proportion for x:
3
2=4
x+ 1.
Solution
Step 1: Cross multiply to eliminate the fractions:
3(x+ 1) = 2 ·4
3x+ 3 = 8
Step 2: Subtract 3 from both sides:
3x= 8 −3
3x= 5
Step 3: Divide by 3 on both sides to solve for x:
x=5
3
Thus, the solution to the proportion is x=5
3.
Question 23
Question
Solve the following proportion for x:3
4=x−1
2x+1 .
Solution
To solve the proportion 3
4=x−1
2x+1 for x, we can cross multiply to eliminate the
fractions.
Step 1: Cross multiply to get rid of the fractions.
3(2x+ 1) = 4(x−1)
Step 2: Expand both sides of the equation.
6x+ 3 = 4x−4
Step 3: Rearrange the equation by isolating the variable terms on one side.
6x−4x=−4−3
14
2x=−7
Step 4: Solve for xby dividing both sides by 2.
x=−7
2
Step 5: Therefore, the solution to the proportion 3
4=x−1
2x+1 is x=−7
2.
Question 24
Question
If 6 men can complete a construction project in 10 days, and 8 women can
complete the same project in 15 days, how many days will it take for 3 men and
4 women working together to complete the project?
Solution
Step 1: Let’s first find the work rates of men and women separately. Let mbe
the work rate of a man and wbe the work rate of a woman. From the given
information, we have: 6 men complete the project in 10 days =⇒6m=1
10 =⇒
m=1
60 . 8 women complete the project in 15 days =⇒8w=1
15 =⇒w=1
120 .
Step 2: Next, let’s find the work rate when 3 men and 4 women work to-
gether. The work rate of 3 men and 4 women working together is (3m+ 4w),
which is equal to 3×1
60 + 4 ×1
120 =1
20 .
Step 3: Finally, let’s find how many days it will take for 3 men and 4 women
to complete the project together. Let dbe the number of days required. By the
work-rate formula Work rate = 1
time , we have
1
20 =1
d=⇒d= 20 days
Therefore, it will take 3 men and 4 women working together 20 days to
complete the project.
Question 25
Question
Solve the following proportion for x:
3
x+ 1 =x+ 2
5
15
Solution
Step 1: Cross multiply to eliminate the fractions.
3·5=(x+ 1)(x+ 2)
Step 2: Simplify and expand the expression.
15 = x2+ 3x+ 2
Step 3: Rearrange the quadratic equation to set it equal to zero.
x2+ 3x+ 2 −15 = 0
Step 4: Combine like terms.
x2+ 3x−13 = 0
Step 5: Solve the quadratic equation using the quadratic formula:
x=−b±√b2−4ac
2a
Step 6: Substitute a= 1, b= 3, and c=−13.
x=−3±p32−4·1·(−13)
2·1
Step 7: Calculate the discriminant under the square root.
x=−3±√9 + 52
2
Step 8: Simplify the expression further.
x=−3±√61
2
Therefore, the solutions for xare x=−3+√61
2and x=−3−√61
2.
Question 26
Question
In a certain company, the ratio of managers to non-managers is 3:7. If there are
100 more non-managers than managers, how many managers are there in the
company?
16
Solution
Let’s denote the number of managers as 3xand the number of non-managers as
7x. We are given that 7x= 3x+ 100. Step 1: Set up the equation based on
the given information.
Since we are given that there are 100 more non-managers than managers, we
have the equation: 7x= 3x+ 100.
Step 2: Solve the equation for x.
Subtracting 3xfrom both sides, we get: 4x= 100.
Dividing by 4, we find: x= 25.
Step 3: Calculate the number of managers.
Now, we can find the number of managers: 3x= 3(25) = 75.
Step 4: Check the answer.
To confirm our solution, we can calculate the number of non-managers: 7x=
7(25) = 175, which is indeed 100 more than the number of managers.
Therefore, there are 75 managers in the company.
Question 27
Question
A group of students decided to share a certain number of pencils. If each student
receives 4 pencils more than they planned, the number of pencils each student
receives will be increased by 40
Solution
Let’s denote the original number of students as xand the original number of
pencils as y.
Step 1: Translate the problem into equations If each student receives
4 pencils more than they planned, the number of pencils each student receives
will be increased by 40
1.4y=x(y+ 4)
If they had originally planned for 12 students more, the number of students
would have been x+ 12. The number of pencils to be shared remains the same,
so we get:
x(y+ 4) = (x+ 12)y
Step 2: Solve the system of equations Now, we solve the system of
equations:
(1.4y=x(y+ 4)
x(y+ 4) = (x+ 12)y
First, we simplify the equations:
1.4y=xy + 4x(1)
17
xy + 4x=xy + 12y(2)
Subtracting equation (1) from equation (2) gives:
4x= 12y=⇒x= 3y
Step 3: Substitute x= 3yinto one of the original equations Substi-
tute x= 3yinto equation (1):
1.4y= 3y(y+ 4)
1.4y= 3y2+ 12y
3y2−1.6y= 0
y(3y−1.6) = 0
Since ycannot be zero, we have:
3y−1.6 = 0 =⇒y=1.6
3=8
15
Step 4: Find the value of xSubstitute y=8
15 back into x= 3y:
x= 3 ×8
15 =8
5= 1.6
Therefore, the initial number of students was 1.6, which is not possible. This
means there was a mistake in the problem setup.
Question 28
Question
Solve the following proportion for xif 2
x+5 =x−1
3.
Solution
Step 1: Cross multiply to eliminate the fractions.
2(3) = (x+ 5)(x−1)
6 = x2+ 4x−5
Step 2: Rearrange the equation into standard form.
x2+ 4x−11 = 0
Step 3: Use the quadratic formula to solve for x.
x=−b±√b2−4ac
2a
18
Step 4: Substitute a= 1, b= 4, and c=−11 into the formula.
x=−4±p42−4(1)(−11)
2(1)
Step 5: Simplify the expression under the square root.
x=−4±√16 + 44
2
x=−4±√60
2
x=−4±2√15
2
Step 6: Simplify the expression by factoring out a 2 from the numerator.
x=2(−2±√15)
2
Step 7: Cancel out the common factor of 2.
x=−2±√15
Therefore, the solutions for xare x=−2 + √15 and x=−2−√15.
Question 29
Question
Solve for x:3
x+ 2 =5
x−3.
Solution
Step 1: Cross multiply to get rid of the fractions.
3(x−3) = 5(x+ 2)
3x−9=5x+ 10
Step 2: Rearrange the equation by isolating xterms on one side.
3x−5x= 10 + 9
−2x= 19
Step 3: Solve for xby dividing both sides by −2.
x=19
−2
x=−19
2
Therefore, the solution to the equation is x=−19
2.
19
Question 30
Question
Solve the following proportion for x:
3
2x−1=5
4x+ 6
Solution
Let’s cross multiply to solve the proportion:
Step 1: Cross multiply to obtain:
3(4x+ 6) = 5(2x−1)
Step 2: Expand both sides of the equation:
12x+ 18 = 10x−5
Step 3: Rearrange the equation by moving all xterms to the left side:
12x−10x=−5−18
2x=−23
Step 4: Finally, divide both sides by 2 to solve for x:
x=−23
2
Therefore, the solution to the proportion is x=−23
2.
Question 31
Question
Solve the following proportion: 5x
6=2x+3
9.
Solution
Step 1: Cross multiply to eliminate the fractions.
5x
6=2x+ 3
9
9·5x= 6 ·(2x+ 3)
Step 2: Simplify both sides of the equation.
45x= 12x+ 18
20
Step 3: Move all terms involving xto one side of the equation.
45x−12x= 18
33x= 18
Step 4: Solve for xby dividing both sides by 33.
x=18
33 =6
11
Therefore, the solution to the proportion is x=6
11 .
Question 32
Question
If 40
Solution
Step 1: Let’s denote the two numbers in the ratio as 2xand 5x.
Step 2: We know that 40
40
100(2x) = 30
100(5x)
Step 3: Simplifying the equation, we get:
0.4(2x)=0.3(5x)
0.8x= 1.5x
Step 4: Solving for x, we find:
1.5x−0.8x= 0
0.7x= 0
x= 0
Step 5: Since x= 0, the numbers are 2(0) = 0 and 5(0) = 0.
Step 6: Therefore, the larger number is 0 .
Question 33
Question
Simplify the following expression: 2x2−6x
4x2−12 .
21
Solution
Step 1: Factor out a 2xfrom the numerator and a 4 from the denominator.
2x2−6x
4x2−12 =2x(x−3)
4(x2−3)
Step 2: Simplify by canceling out common factors.
2x(x−3)
4(x2−3) =2(x−3)
2(x2−3) =x−3
x2−3
Therefore, the simplified expression is x−3
x2−3.
Question 34
Question
If a certain quantity is divided into three parts that are in the ratio 3 : 4 : 5,
and the smallest part is 12, what is the quantity?
Solution
Let’s denote the quantity as Q. Since the parts are in the ratio 3 : 4 : 5, we can
express the parts as 3x, 4x, and 5x, where xis a constant of proportionality.
Step 1: Write the equation based on the information given. From the
problem, we know that the smallest part is 12. This means that 3x= 12. We
can solve for xto find the value of each part.
3x= 12
x= 4
Step 2: Find the value of the quantity. Now that we know the value of x,
we can find the three parts:
3x= 3(4) = 12
4x= 4(4) = 16
5x= 5(4) = 20
The total quantity, Q, is the sum of these three parts:
Q= 3x+ 4x+ 5x
Q= 12 + 16 + 20
Q= 48
Therefore, the quantity is 48.
22
Question 35
Question
A recipe for chocolate chip cookies requires 1 1/2 cups of flour for every 1 cup of
sugar. If you want to make a batch of cookies using 3 cups of flour, how many
cups of sugar should you use?
Solution
Let xbe the number of cups of sugar needed to make the batch of cookies using
3 cups of flour.
Step 1: Set up a proportion using the given ratio:
11
2
1=3
x
Step 2: Simplify the proportion:
3/2
1=3
x
3
2=3
x
Step 3: Cross multiply to solve for x:
3x= 2 ×3
3x= 6
Step 4: Divide by 3 to solve for x:
x=6
3
x= 2
Therefore, you should use 2 cups of sugar to make a batch of cookies using
3 cups of flour.
23
Question 2
Question
Solve the following proportion:
3x+ 2
4=5x−1
6
Solution
Step 1: Cross multiply to eliminate the fractions:
6(3x+ 2) = 4(5x−1)
18x+ 12 = 20x−4
Step 2: Rearrange the equation by moving all terms involving xto one side:
18x+ 12 = 20x−4
18x−20x=−4−12
−2x=−16
Step 3: Solve for x:
−2x=−16
x=−16
−2
x= 8
Therefore, the solution to the proportion is x= 8.
Question 3
Question
Solve the following proportion: 2x+3
5x−1=4x−1
3x+2 .
Solution
To solve the proportion, we will cross multiply and then solve for x.
Step 1: Cross multiply to get: (2x+ 3)(3x+ 2) = (5x−1)(4x−1).
Step 2: Expand both sides: 6x2+ 4x+ 9x+ 6 = 20x2−5x−4x+ 1
Step 3: Combine like terms on both sides: 6x2+ 13x+ 6 = 20x2−9x+ 1
Step 4: Subtract 6x2+ 13x+ 6 from both sides: 0 = 14x2−22x−5
Step 5: Factor the quadratic equation: (2x+ 1)(7x−5) = 0
Step 6: Set each factor equal to zero and solve for x: 2x+1 = 0 or 7x−5=0
Step 7: Solve for x: For 2x+ 1 = 0, we get 2x=−1
⇒x=−1
2.
2
For 7x−5 = 0, we get 7x= 5
⇒x=5
7.
Therefore, the solutions are x=−1
2and x=5
7.
Question 4
Question
If 5 pencils and 4 pens cost $24.00, and 3 pencils and 2 pens cost $12.80, what
is the cost of 1 pencil?
Solution
Step 1: Set up a system of equations based on the given information. Let xbe
the cost of 1 pencil and ybe the cost of 1 pen. Then we can write:
(5x+ 4y= 24.00
3x+ 2y= 12.80
Step 2: Solve the system of equations. We can solve this system using the
method of substitution or elimination. Let’s use substitution in this case. From
the second equation, we can solve for yto get:
y=12.80 −3x
2
Now substitute yinto the first equation:
5x+ 4 12.80 −3x
2= 24.00
Simplify the equation:
5x+ 4(6.40 −1.5x) = 24.00
5x+ 25.60 −6x= 24.00
−x=−1.6
x= 1.6
Step 3: Find the cost of 1 pencil. Therefore, the cost of 1 pencil is $1.60.
Question 5
Question
If 5 men can complete a construction project in 10 days, and 8 women can
complete the same project in 15 days, how many days will it take 3 men and 4
women working together to complete the project?
3
Solution
Let’s first find the rate at which each group works, in terms of fraction of the
project completed per day. Then we will use these rates to find the time it takes
for the combined group to finish the project.
Step 1: Find the rate at which one man works. Let xbe the fraction of the
project that one man can complete in one day. Given that 5 men can complete
the project in 10 days, we have:
5 men ×10 days = 1 project ⇒50x= 1 ⇒x=1
50
Therefore, one man can complete 1
50 of the project in one day.
Step 2: Find the rate at which one woman works. Let ybe the fraction of
the project that one woman can complete in one day. Given that 8 women can
complete the project in 15 days, we have:
8 women ×15 days = 1 project ⇒120y= 1 ⇒y=1
120
Therefore, one woman can complete 1
120 of the project in one day.
Step 3: Find the combined rate of 3 men and 4 women. The combined rate
is the sum of the rates of the men and women working together:
31
50+ 4 1
120=3
50 +1
30 =9+5
150 =14
150 =7
75
Therefore, the combined group can complete 7
75 of the project in one day.
Step 4: Find the time it takes for 3 men and 4 women to complete the
project. Let dbe the number of days it takes for 3 men and 4 women to
complete the project. Using the combined rate, we have:
d×7
75 = 1 ⇒d=75
7≈10.71 days
Therefore, it will take approximately 10.71 days for 3 men and 4 women
working together to complete the project.
Question 6
Question
If 5 men can complete a construction project in 10 days, how many days will it
take for 8 men to complete the same project?
Solution
Let’s denote the number of days it takes for 8 men to complete the project as
d. We can set up a proportion based on the information given:
4
5 men ×10 days
8 men ×ddays = 1
We can solve this proportion for d.
50
8d= 1
Step 1: Multiply both sides by 8dto isolate d.
50 = 8d
Step 2: Divide both sides by 8 to solve for d.
d=50
8= 6.25
Therefore, it will take 8 men approximately 6.25 days to complete the con-
struction project.
Question 7
Question
A recipe for a cake calls for 2 cups of flour and 3 eggs. If you want to make
4 cakes, how many cups of flour will you need if the number of eggs used is
doubled?
Solution
Let xrepresent the number of cups of flour needed for 4 cakes when the number
of eggs used is doubled.
Step 1: Determine the initial ratio of flour to eggs. For 1 cake: - Cups of
flour: 2 cups - Number of eggs: 3 eggs Thus, the ratio of cups of flour to eggs
is 2 : 3.
Step 2: Determine the new ratio of flour to eggs when the number of eggs
used is doubled. Since the number of eggs is doubled, the new ratio becomes: -
Cups of flour: xcups - Number of eggs: 3 ×2 = 6 eggs Hence, the new ratio of
cups of flour to eggs is x: 6.
Step 3: Set up a proportion to find the value of x:
2
3=x
6
Step 4: Solve for x:
2×6=3x
12 = 3x
x=12
3
5
x= 4
Therefore, you will need 4 cups of flour to make 4 cakes when the number
of eggs used is doubled.
Question 8
Question
In a bag of marbles, the ratio of red marbles to blue marbles is 3:4. If there are
28 red marbles, how many blue marbles are in the bag?
Solution
Let’s set up a proportion to find the number of blue marbles in the bag.
Step 1: Determine the ratio of red marbles to blue marbles The ratio of red
marbles to blue marbles is given as 3:4.
Step 2: Write the proportion using the given ratio and number of red mar-
bles Let xrepresent the number of blue marbles. We can set up the proportion:
3
4=28
x.
Step 3: Cross multiply to solve for xCross multiplying the proportion, we
get 3x= 4 ∗28.
Step 4: Solve for xSolving the equation, we find: 3x= 112
x=112
3
x= 371
3.
Thus, there are 37 blue marbles in the bag.
Question 9
Question
In a research study, the ratio of 1st-year students to 2nd-year students is 5:3. If
there are 240 1st-year students, how many 2nd-year students are there?
Solution
Step 1: Let xrepresent the number of 2nd-year students. Step 2: Use the given
ratio to set up the proportion: 240
x=5
3. Step 3: Cross multiply to solve for x:
240 ×3=5x. Step 4: Simplify the equation: 720 = 5x. Step 5: Solve for x:
x=720
5= 144. Step 6: There are 144 2nd-year students in the research study.
Question 10
Question
Solve for xin the proportion: 2
3=x+1
4x−3.
6
Solution
To solve for xin the proportion 2
3=x+1
4x−3, we will cross multiply.
Step 1: Cross multiply to get rid of the fractions.
2(4x−3) = 3(x+ 1)
8x−6 = 3x+ 3
Step 2: Rearrange the equation to isolate x.
8x−6=3x+ 3
8x−3x= 6 + 3
5x= 9
Step 3: Solve for xby dividing both sides by 5.
x=9
5
Therefore, the solution to the proportion is x=9
5.
Question 11
Question
Solve the following proportion for x:
3x+ 4
5=2x−1
3
Solution
To solve the proportion for x, we need to cross multiply and solve the resulting
equation.
Step 1: Cross multiply to get rid of the fractions:
(3x+ 4) ×3=5×(2x−1)
Step 2: Expand both sides of the equation:
9x+ 12 = 10x−5
Step 3: Rearrange the equation to isolate xon one side:
9x+ 12 = 10x−5
12 + 5 = 10x−9x
17 = x
Step 4: Therefore, the solution to the proportion is x= 17.
7
Question 12
Question
A recipe calls for 2 cups of sugar for every 3 cups of flour. If you want to make a
bigger batch of the recipe using 12 cups of flour, how many cups of sugar should
you use?
Solution
Step 1: Find the ratio of sugar to flour in the original recipe. Let xrepresent
the number of cups of sugar needed for 3 cups of flour. The ratio of sugar to
flour in the original recipe is 2
3, which means x
3=2
3. Solving for xgives x= 2.
Step 2: Determine the number of cups of sugar needed for 12 cups of flour.
Since for 3 cups of flour you need 2 cups of sugar, for 12 cups of flour you will
need 2
3×12 = 8 cups of sugar.
Therefore, you should use 8 cups of sugar when making a bigger batch of
the recipe with 12 cups of flour.
Question 13
Question
A recipe for a cake requires 2 cups of flour for every 3 eggs. If you want to make
a cake that uses 6 cups of flour, how many eggs should you use?
Solution
Step 1: Let’s set up a proportion using the given information. Let xrepresent
the number of eggs needed.
2 cups of flour
3 eggs =6 cups of flour
xeggs
Step 2: Cross multiply to solve for x.
2×x= 6 ×3
Step 3: Simplify the equation.
2x= 18
Step 4: Divide both sides by 2 to find the value of x.
x=18
2= 9
Therefore, you should use 9 eggs to make a cake that uses 6 cups of flour.
8
Question 14
Question
A recipe for banana bread calls for 2 cups of flour for every 3 bananas. If you
only have 7 bananas, how many cups of flour should you use to maintain the
same ratio?
Solution
Let the unknown number of cups of flour be represented by x. We can set up a
proportion based on the relationship between the cups of flour and the number
of bananas:
2
3=x
7
Step 1: Cross multiply to solve for x:
2×7=3x
14 = 3x
Step 2: Divide both sides by 3 to solve for x:
x=14
3= 42
3cups
Therefore, you should use 42
3cups of flour if you only have 7 bananas in
order to maintain the same ratio as the original recipe.
Question 15
Question
If a:b= 2 : 3 and b:c= 4 : 5, find a:b:c.
Solution
Step 1: We are given that a:b= 2 : 3 and b:c= 4 : 5.
Let’s represent these ratios using variables and constants:
a= 2x, b = 3x= 4y, c = 5y
where xand yare constants.
Step 2: To find the value of xand y, equate the expression for b:
3x= 4y
9
x=4
3y
Step 3: Since a= 2x, substitute the expression for xfound in Step 2:
a= 2 4
3y=8
3y
Step 4: Now we have the values of a,b, and cin terms of y. So the ratio
a:b:ccan be written as:
8
3y: 3x: 5y=8
3y: 3 4
3y: 5y=8:4:5
Step 5: Therefore, the ratio a:b:cis 8 : 4 : 5.
Question 16
Question
If 24 pounds of apples cost 36, howmuchdo48poundsofapplescost?
Solution
Let xrepresent the cost of 48 pounds of apples.
Step 1: Set up a ratio using the given information. The cost is directly
proportional to the weight of apples, so we can write the ratio:
24
36 =48
x
Step 2: Solve for xby cross multiplying:
24x= 36 ×48
24x= 1728
Step 3: Divide by 24 to solve for x:
x=1728
24
x= 72
Therefore, 48 pounds of apples cost 72.
Question 17
Question
Solve for xin the following proportion: 2x
3=x−1
2.
10
Solution
To solve this proportion, we will cross multiply and solve for x.
Step 1: Cross multiply the terms in the proportion.
2x·2=3·(x−1)
4x= 3x−3
Step 2: Subtract 3xfrom both sides of the equation.
4x−3x= 3x−3−3x
x=−3
Step 3: Check the solution by substituting x=−3 back into the original
proportion.
2(−3)
3=(−3) −1
2
−6
3=−4
2
−2 = −2
Therefore, the solution to the proportion is x=−3.
Question 18
Question
Solve for xin the following proportion:
3x+ 5
4x−1=7
9
Solution
Step 1: Cross multiply to get rid of the fractions.
9(3x+ 5) = 7(4x−1)
27x+ 45 = 28x−7
Step 2: Simplify the equation.
27x+ 45 = 28x−7
45 + 7 = 28x−27x
52 = x
Conclusion: The solution to the given proportion is x= 52.
11
Question 19
Question
Given that a:b= 3 : 2 and b:c= 4 : 5, find the ratio a:b:c.
Solution
We are given that a:b= 3 : 2 and b:c= 4 : 5. To find the ratio a:b:c, we
need to express a,b, and cin terms of the given ratios and then combine them.
Step 1: Find the value of bSince a:b= 3 : 2 and b:c= 4 : 5, we can
equate the second terms to find the value of b:
a
b=3
2and b
c=4
5
Cross multiplying gives us:
2a= 3band 4c= 5b
Solving these two equations simultaneously, we find:
b=2
3a=4
5c
Step 2: Express a,b, and cin terms of the common factor Let’s
express aand cin terms of b:
a=3
2b
c=5
4b
Step 3: Write the ratio a:b:cNow that we have expressions for a,b,
and cin terms of b, we can write the ratio a:b:cas:
a:b:c=3
2b:b:5
4b=3:3
2:5=6:3:10
Therefore, the ratio a:b:cis 6 : 3 : 10 .
Question 20
Question
Solve for x:3x
4=5
7.
12
Solution
Step 1: Cross multiply to eliminate the fractions.
3x
4=5
7
3x×7=4×5
Step 2: Simplify the expression.
21x= 20
Step 3: Solve for xby dividing both sides by 21.
x=20
21
Therefore, the solution is x=20
21 .
Question 21
Question
In a survey, it was found that 45
Solution
Let xbe the number of respondents who preferred brand A.
Step 1: Write the proportion using the given information:
x
630 =45
100
Step 2: Simplify the proportion by dividing both sides by 100:
x
630 = 0.45
Step 3: Solve for xby multiplying both sides by 630:
x= 0.45 ×630
Step 4: Calculate the number of respondents who preferred brand A:
x= 283.5
Step 5: Since we cannot have a fraction of a respondent, round up to the
nearest whole number:
x≈284
Therefore, 284 respondents preferred brand A out of the 630 total respon-
dents.
13
Question 22
Question
Solve the following proportion for x:
3
2=4
x+ 1.
Solution
Step 1: Cross multiply to eliminate the fractions:
3(x+ 1) = 2 ·4
3x+ 3 = 8
Step 2: Subtract 3 from both sides:
3x= 8 −3
3x= 5
Step 3: Divide by 3 on both sides to solve for x:
x=5
3
Thus, the solution to the proportion is x=5
3.
Question 23
Question
Solve the following proportion for x:3
4=x−1
2x+1 .
Solution
To solve the proportion 3
4=x−1
2x+1 for x, we can cross multiply to eliminate the
fractions.
Step 1: Cross multiply to get rid of the fractions.
3(2x+ 1) = 4(x−1)
Step 2: Expand both sides of the equation.
6x+ 3 = 4x−4
Step 3: Rearrange the equation by isolating the variable terms on one side.
6x−4x=−4−3
14
2x=−7
Step 4: Solve for xby dividing both sides by 2.
x=−7
2
Step 5: Therefore, the solution to the proportion 3
4=x−1
2x+1 is x=−7
2.
Question 24
Question
If 6 men can complete a construction project in 10 days, and 8 women can
complete the same project in 15 days, how many days will it take for 3 men and
4 women working together to complete the project?
Solution
Step 1: Let’s first find the work rates of men and women separately. Let mbe
the work rate of a man and wbe the work rate of a woman. From the given
information, we have: 6 men complete the project in 10 days =⇒6m=1
10 =⇒
m=1
60 . 8 women complete the project in 15 days =⇒8w=1
15 =⇒w=1
120 .
Step 2: Next, let’s find the work rate when 3 men and 4 women work to-
gether. The work rate of 3 men and 4 women working together is (3m+ 4w),
which is equal to 3×1
60 + 4 ×1
120 =1
20 .
Step 3: Finally, let’s find how many days it will take for 3 men and 4 women
to complete the project together. Let dbe the number of days required. By the
work-rate formula Work rate = 1
time , we have
1
20 =1
d=⇒d= 20 days
Therefore, it will take 3 men and 4 women working together 20 days to
complete the project.
Question 25
Question
Solve the following proportion for x:
3
x+ 1 =x+ 2
5
15
Solution
Step 1: Cross multiply to eliminate the fractions.
3·5=(x+ 1)(x+ 2)
Step 2: Simplify and expand the expression.
15 = x2+ 3x+ 2
Step 3: Rearrange the quadratic equation to set it equal to zero.
x2+ 3x+ 2 −15 = 0
Step 4: Combine like terms.
x2+ 3x−13 = 0
Step 5: Solve the quadratic equation using the quadratic formula:
x=−b±√b2−4ac
2a
Step 6: Substitute a= 1, b= 3, and c=−13.
x=−3±p32−4·1·(−13)
2·1
Step 7: Calculate the discriminant under the square root.
x=−3±√9 + 52
2
Step 8: Simplify the expression further.
x=−3±√61
2
Therefore, the solutions for xare x=−3+√61
2and x=−3−√61
2.
Question 26
Question
In a certain company, the ratio of managers to non-managers is 3:7. If there are
100 more non-managers than managers, how many managers are there in the
company?
16
Solution
Let’s denote the number of managers as 3xand the number of non-managers as
7x. We are given that 7x= 3x+ 100. Step 1: Set up the equation based on
the given information.
Since we are given that there are 100 more non-managers than managers, we
have the equation: 7x= 3x+ 100.
Step 2: Solve the equation for x.
Subtracting 3xfrom both sides, we get: 4x= 100.
Dividing by 4, we find: x= 25.
Step 3: Calculate the number of managers.
Now, we can find the number of managers: 3x= 3(25) = 75.
Step 4: Check the answer.
To confirm our solution, we can calculate the number of non-managers: 7x=
7(25) = 175, which is indeed 100 more than the number of managers.
Therefore, there are 75 managers in the company.
Question 27
Question
A group of students decided to share a certain number of pencils. If each student
receives 4 pencils more than they planned, the number of pencils each student
receives will be increased by 40
Solution
Let’s denote the original number of students as xand the original number of
pencils as y.
Step 1: Translate the problem into equations If each student receives
4 pencils more than they planned, the number of pencils each student receives
will be increased by 40
1.4y=x(y+ 4)
If they had originally planned for 12 students more, the number of students
would have been x+ 12. The number of pencils to be shared remains the same,
so we get:
x(y+ 4) = (x+ 12)y
Step 2: Solve the system of equations Now, we solve the system of
equations:
(1.4y=x(y+ 4)
x(y+ 4) = (x+ 12)y
First, we simplify the equations:
1.4y=xy + 4x(1)
17
xy + 4x=xy + 12y(2)
Subtracting equation (1) from equation (2) gives:
4x= 12y=⇒x= 3y
Step 3: Substitute x= 3yinto one of the original equations Substi-
tute x= 3yinto equation (1):
1.4y= 3y(y+ 4)
1.4y= 3y2+ 12y
3y2−1.6y= 0
y(3y−1.6) = 0
Since ycannot be zero, we have:
3y−1.6 = 0 =⇒y=1.6
3=8
15
Step 4: Find the value of xSubstitute y=8
15 back into x= 3y:
x= 3 ×8
15 =8
5= 1.6
Therefore, the initial number of students was 1.6, which is not possible. This
means there was a mistake in the problem setup.
Question 28
Question
Solve the following proportion for xif 2
x+5 =x−1
3.
Solution
Step 1: Cross multiply to eliminate the fractions.
2(3) = (x+ 5)(x−1)
6 = x2+ 4x−5
Step 2: Rearrange the equation into standard form.
x2+ 4x−11 = 0
Step 3: Use the quadratic formula to solve for x.
x=−b±√b2−4ac
2a
18
Step 4: Substitute a= 1, b= 4, and c=−11 into the formula.
x=−4±p42−4(1)(−11)
2(1)
Step 5: Simplify the expression under the square root.
x=−4±√16 + 44
2
x=−4±√60
2
x=−4±2√15
2
Step 6: Simplify the expression by factoring out a 2 from the numerator.
x=2(−2±√15)
2
Step 7: Cancel out the common factor of 2.
x=−2±√15
Therefore, the solutions for xare x=−2 + √15 and x=−2−√15.
Question 29
Question
Solve for x:3
x+ 2 =5
x−3.
Solution
Step 1: Cross multiply to get rid of the fractions.
3(x−3) = 5(x+ 2)
3x−9=5x+ 10
Step 2: Rearrange the equation by isolating xterms on one side.
3x−5x= 10 + 9
−2x= 19
Step 3: Solve for xby dividing both sides by −2.
x=19
−2
x=−19
2
Therefore, the solution to the equation is x=−19
2.
19
Question 30
Question
Solve the following proportion for x:
3
2x−1=5
4x+ 6
Solution
Let’s cross multiply to solve the proportion:
Step 1: Cross multiply to obtain:
3(4x+ 6) = 5(2x−1)
Step 2: Expand both sides of the equation:
12x+ 18 = 10x−5
Step 3: Rearrange the equation by moving all xterms to the left side:
12x−10x=−5−18
2x=−23
Step 4: Finally, divide both sides by 2 to solve for x:
x=−23
2
Therefore, the solution to the proportion is x=−23
2.
Question 31
Question
Solve the following proportion: 5x
6=2x+3
9.
Solution
Step 1: Cross multiply to eliminate the fractions.
5x
6=2x+ 3
9
9·5x= 6 ·(2x+ 3)
Step 2: Simplify both sides of the equation.
45x= 12x+ 18
20
Step 3: Move all terms involving xto one side of the equation.
45x−12x= 18
33x= 18
Step 4: Solve for xby dividing both sides by 33.
x=18
33 =6
11
Therefore, the solution to the proportion is x=6
11 .
Question 32
Question
If 40
Solution
Step 1: Let’s denote the two numbers in the ratio as 2xand 5x.
Step 2: We know that 40
40
100(2x) = 30
100(5x)
Step 3: Simplifying the equation, we get:
0.4(2x)=0.3(5x)
0.8x= 1.5x
Step 4: Solving for x, we find:
1.5x−0.8x= 0
0.7x= 0
x= 0
Step 5: Since x= 0, the numbers are 2(0) = 0 and 5(0) = 0.
Step 6: Therefore, the larger number is 0 .
Question 33
Question
Simplify the following expression: 2x2−6x
4x2−12 .
21
Solution
Step 1: Factor out a 2xfrom the numerator and a 4 from the denominator.
2x2−6x
4x2−12 =2x(x−3)
4(x2−3)
Step 2: Simplify by canceling out common factors.
2x(x−3)
4(x2−3) =2(x−3)
2(x2−3) =x−3
x2−3
Therefore, the simplified expression is x−3
x2−3.
Question 34
Question
If a certain quantity is divided into three parts that are in the ratio 3 : 4 : 5,
and the smallest part is 12, what is the quantity?
Solution
Let’s denote the quantity as Q. Since the parts are in the ratio 3 : 4 : 5, we can
express the parts as 3x, 4x, and 5x, where xis a constant of proportionality.
Step 1: Write the equation based on the information given. From the
problem, we know that the smallest part is 12. This means that 3x= 12. We
can solve for xto find the value of each part.
3x= 12
x= 4
Step 2: Find the value of the quantity. Now that we know the value of x,
we can find the three parts:
3x= 3(4) = 12
4x= 4(4) = 16
5x= 5(4) = 20
The total quantity, Q, is the sum of these three parts:
Q= 3x+ 4x+ 5x
Q= 12 + 16 + 20
Q= 48
Therefore, the quantity is 48.
22
Question 35
Question
A recipe for chocolate chip cookies requires 1 1/2 cups of flour for every 1 cup of
sugar. If you want to make a batch of cookies using 3 cups of flour, how many
cups of sugar should you use?
Solution
Let xbe the number of cups of sugar needed to make the batch of cookies using
3 cups of flour.
Step 1: Set up a proportion using the given ratio:
11
2
1=3
x
Step 2: Simplify the proportion:
3/2
1=3
x
3
2=3
x
Step 3: Cross multiply to solve for x:
3x= 2 ×3
3x= 6
Step 4: Divide by 3 to solve for x:
x=6
3
x= 2
Therefore, you should use 2 cups of sugar to make a batch of cookies using
3 cups of flour.
23
