MATH 114 - QUANTITATIVE
REASONING - Ratios and Proportions
Question Bank - Set 3
Liberty University
Question 1
Question
In a mixture of alcohol and water, the ratio of alcohol to water is 5:3. If 8 liters
of water is added to the mixture, the ratio becomes 5:4. Find the amount of
alcohol in the mixture initially.
Solution
Let the amount of alcohol and water originally be 5xand 3xliters, respectively.
Step 1: Set up the initial ratio equation: Initially, the ratio of alcohol to
water is 5x
3x. This can be simplified to 5
3.
Step 2: Set up the final ratio equation: After 8 liters of water is added, the
amount of water becomes 3x+ 8 liters. The ratio of alcohol to water becomes
5x
3x+8 . This ratio is given as 5
4.
Step 3: Solve for x:5x
3x+ 8 =5
4
20x= 15x+ 40
5x= 40
x= 8
Step 4: Find the amount of alcohol initially: The amount of alcohol initially
is 5x= 5(8) = 40 liters.
Question 2
Question
Simplify the following ratio: 4x2y
12xy3.
Solution
Step 1: Simplify the numerator and denominator separately. Step 2: Divide the
numerator by the denominator.
Step 1: To simplify the numerator 4x2y, we rewrite it as 4 ·x·x·y.
To simplify the denominator 12xy3, we rewrite it as 12 ·x·y·y·y.
Step 2: Dividing the numerator by the denominator:
4x2y
12xy3=4·x·x·y
12 ·x·y·y·y=1
3·x
y2=x
3y2
Question 3
Question
Solve the following proportion for x:
2x+ 1
3x−2=3x+ 5
4x−1
Solution
To solve the given proportion for x, we will first cross multiply and then simplify
the resulting equation step by step.
Step 1: Cross multiply to eliminate the denominators:
(2x+ 1)(4x−1) = (3x−2)(3x+ 5)
Step 2: Expand both sides of the equation:
8x2−2x+ 4x−1=9x2+ 15x−6x−10
Step 3: Simplify both sides of the equation:
8x2+ 2x−1=9x2+ 9x−10
Step 4: Rearrange the equation to set it equal to zero:
0 = x2+ 7x−9
Step 5: Solve the quadratic equation by factoring or using the quadratic
formula:
0=(x+ 9)(x−1)
Step 6: Set each factor to zero and solve for x:
x+ 9 = 0 =⇒x=−9 or x−1 = 0 =⇒x= 1
2
Step 7: Check for extraneous solutions by substituting each back into the
original proportion: Checking x=−9:
2(−9) + 1
3(−9) −2=3(−9) + 5
4(−9) −1
−17
−29 =−22
−37
17
29 =22
37
Checking x= 1:
2(1) + 1
3(1) −2=3(1) + 5
4(1) −1
3
1=8
3
3 = 22
3
Therefore, the solution to the proportion is x= 1.
Question 4
Question
Solve for xgiven that 3x+1
5x−2=5
9.
Solution
Step 1: Cross multiply to get rid of the fractions.
3x+ 1
5x−2=5
9
9(3x+ 1) = 5(5x−2)
Step 2: Expand both sides of the equation.
27x+ 9 = 25x−10
Step 3: Simplify the equation by collecting like terms.
27x+ 9 = 25x−10
27x−25x=−10 −9
2x=−19
Step 4: Solve for xby dividing both sides by 2.
2x=−19
x=−19
2
Therefore, the solution is x=−19
2.
3
Question 5
Question
A construction crew is building a scale model of a house. If 2.5 feet of a wall of
the actual house corresponds to 0.1 inches on the model, how many inches on
the model represent 9 feet of the actual wall?
Solution
Step 1: Calculate the scale factor. Let xbe the number of inches on the model
that represent 9 feet of the actual wall. We can set up a proportion to find the
scale factor: 2.5 feet
0.1 inches =9 feet
xinches
Step 2: Solve for x. Cross multiply to solve for x:
2.5×x= 0.1×9
2.5x= 0.9
Step 3: Calculate the value of x. Divide by 2.5 to find the value of x:
x=0.9
2.5
x= 0.36
Therefore, 0.36 inches on the model represent 9 feet of the actual wall.
Question 6
Question
Solve the following proportion for x:
2x+ 3
5=4x+ 7
9
Solution
Step 1: Cross multiply to eliminate the fractions:
9(2x+ 3) = 5(4x+ 7)
18x+ 27 = 20x+ 35
Step 2: Rearrange the equation by isolating xterms on one side:
18x−20x= 35 −27
−2x= 8
4
Step 3: Divide by the coefficient of xto solve for x:
x=8
−2
x=−4
Therefore, the solution to the proportion is x=−4.
Question 7
Question
A recipe requires 3 cups of flour for every 2 cups of sugar. If you have 8 cups of
flour, how many cups of sugar do you need?
Solution
Step 1: Calculate the ratio of flour to sugar in the recipe. Let xrepresent the
number of cups of sugar needed. The ratio of flour to sugar in the recipe is 3
2.
This means that 8
x=3
2.
Step 2: Solve for x. Cross multiply to solve for x: 2 ∗8=3x
16 = 3x
x=16
3
x= 51
3
Step 3: Interpret the solution. You need 16
3or 51
3cups of sugar to go with
8 cups of flour in the recipe.
Question 8
Question
Solve the proportion: 2x+1
3=4x−5
5.
Solution
To solve the proportion 2x+1
3=4x−5
5, we will cross-multiply to eliminate the
fractions and then solve for x.
Step 1: Cross-multiply to get: (2x+ 1) ×5=3×(4x−5).
Step 2: Expand both sides: 10x+ 5 = 12x−15.
Step 3: Rearrange the equation to isolate xterms: 10x+ 5 = 12x−15 ⇒
10x+ 15 = 12x−5.
Step 4: Move all terms involving xto one side: 10x−12x=−5−15 ⇒
−2x=−20.
Step 5: Solve for x:−2x=−20 ⇒x=−20
−2⇒x= 10.
Step 6: Check the solution by substituting x= 10 back into the original
proportion.
5
2(10)+1
3=4(10)−5
5
20+1
3=40−5
5
21
3=35
5
7 = 7.
Since the equation holds true, x= 10 is the correct solution.
Question 9
Question
Solve the following proportion for x:
3
x+ 2 =x+ 1
5
Solution
Step 1: Cross multiply to get rid of the fractions.
3·5=(x+ 1)(x+ 2)
Step 2: Simplify the equation.
15 = x2+ 3x+ 2
Step 3: Rearrange the equation into standard quadratic form.
x2+ 3x−13 = 0
Step 4: Use the quadratic formula to solve for x:
x=−b±√b2−4ac
2a
Step 5: Plug in a= 1, b= 3, and c=−13 into the quadratic formula.
x=−3±p32−4(1)(−13)
2(1)
Step 6: Simplify under the square root.
x=−3±√9 + 52
2
x=−3±√61
2
Therefore, the solutions for xare
x=−3 + √61
2and x=−3−√61
2
6
Question 10
Question
Simplify the following expression to its simplest form: 3x2+ 2xy
5x2−3xy .
Solution
Step 1: Factor out xfrom both the numerator and denominator:
x(3x+ 2y)
x(5x−3y)
Step 2: Simplify by canceling out the common factors.
3x+ 2y
5x−3y
Therefore, the simplified form of the expression is 3x+ 2y
5x−3y.
Question 11
Question
Given that xvaries directly with yand inversely with z, and that x= 6 when
y= 3 and z= 4, find xwhen y= 10 and z= 6.
Solution
Step 1: First, we express the direct and inverse variation relationships using the
proportionality constants: Let k1be the constant of direct variation and k2be
the constant of inverse variation. Then, we have the relationships:
x=k1y
x=k2
z
Step 2: Next, we use the given information to find the values of k1and k2:
When x= 6, y= 3, and z= 4, we have:
6 = k1·3
6 = k2
4
Solving these equations gives k1= 2 and k2= 24.
Step 3: Now that we have the proportionality constants, we can find the
value of xwhen y= 10 and z=6:x= 2 ·10 = 20
x=24
6= 4
Therefore, when y= 10 and z= 6, x= 20 or x= 4.
7
Question 12
Question
Solve the following proportion for x:
2x+ 4
3=x+ 5
6
Solution
Step 1: Cross multiply to eliminate the fractions:
6(2x+ 4) = 3(x+ 5)
12x+ 24 = 3x+ 15
Step 2: Simplify the equation:
12x−3x= 15 −24
9x=−9
Step 3: Solve for x:
x=−9
9
x=−1
Thus, the solution to the proportion is x=−1.
Question 13
Question
Samantha and Emily decide to split a sum of money in the ratio 5:3. If Samantha
receives 120morethanEmily, howmuchmoneydideachreceive?
Solution
Step 1: Let’s denote the amount of money that Samantha and Emily re-
ceived as 5xand 3x, respectively. We are also given that Samantha received
120morethanEmily, sowecansetuptheequation : 5x= 3x+ 120
Step 2: Now, we can solve for x:
5x= 3x+ 120
2x= 120
x= 60
8
Step 3: Now that we have found x, we can find out how much money each
person received:
Samantha received 5x= 5(60) = $300
Emily received 3x= 3(60) = $180
Therefore, Samantha received
$
300 and Emily received
$
180.
Question 14
Question
If 3 liters of a solution contains 20
Solution
Step 1: Determine the amount of salt in the initial solution.
Let xrepresent the amount of salt in the 3 liters of 20
0.20 ×3 = x
x= 0.6 liters
Step 2: Set up the equation for the final solution.
Let ybe the amount of pure salt added to the solution. The total volume of
the final solution is 3 + yliters. We want the final solution to be 25
x+y
3 + y= 0.25
Substitute the value of xwe found in Step 1:
0.6 + y
3 + y= 0.25
Step 3: Solve for y.
Simplify the equation:
0.6 + y= 0.25(3 + y)
0.6 + y= 0.75 + 0.25y
0.75 −0.6=0.25y−y
0.15 = −0.75y
y=−0.15
0.75 =−0.2
Step 4: Interpret the solution.
Since the volume of salt cannot be negative, we discard the negative solution.
Therefore, we do not need to add any salt to make the solution 25
9
Question 15
Question
A recipe for bread requires 3 cups of flour, 1 cup of sugar, and 1
2cup of oil.
If you want to make 2 loaves of bread, how much flour, sugar, and oil do you
need?
Solution
Let’s first determine the ratio of ingredients needed to make one loaf of bread:
- Flour: 3 cups - Sugar: 1 cup - Oil: 1
2cup
To make 2 loaves of bread, we simply double the amount of each ingredient:
- Flour: 3 cups×2 = 6 cups - Sugar: 1 cup×2 = 2 cups - Oil: 1
2cup×2 = 1 cup
Therefore, to make 2 loaves of bread, you will need 6 cups of flour, 2 cups
of sugar, and 1 cup of oil.
Question 16
Question
If x,y, and zare positive numbers such that x
y=3
4and y
z=5
6, find the ratio
x
z.
Solution
Step 1: From x
y=3
4, we can rewrite it as x=3
4y.
Step 2: From y
z=5
6, we can rewrite it as y=5
6z.
Step 3: Substituting the expression for yfrom Step 2 into the expression for
xfrom Step 1, we get:
x=3
45
6z=5
8z
Step 4: Therefore, the ratio x
zis:
x
z=
5
8z
z=5
8= 5 : 8
Hence, the ratio x
zis 5 : 8.
Question 17
Question
If xand yare positive real numbers such that x:y= 4 : 3 and x+5
y−2=7
5, find
the value of x.
10
Solution
Step 1: We start by setting up the proportion based on the given ratio x:y=4:
3. Step 2: This means x
y=4
3. We can rewrite this as y=3
4x. Step 3: Substitute
y=3
4xinto the second equation x+5
y−2=7
5. Step 4: We get x+5
(3
4x)−2=7
5. Step
5: Simplifying the denominator, we have x+5
3x
4−2=7
5. Step 6: To eliminate the
fractions, we can multiply both sides by 20 (the least common multiple of 4 and
5). Step 7: This gives 20(x+ 5) = 4(3x)−8. Step 8: Simplifying the equation
gives 20x+ 100 = 12x−8. Step 9: Rearranging terms, we have 8x=−108.
Step 10: Finally, solving for xgives x=−108
8=−13.5. Therefore, the value of
xis −13.5.
Question 18
Question
In a certain company, the ratio of the number of male employees to the number
of female employees is 3:5. If there are 360 employees in total, how many of
them are male?
Solution
Let the number of male employees be 3xand the number of female employees
be 5x. We are given that the total number of employees is 360. Therefore, we
have the equation:
3x+ 5x= 360
Step 1: Combine like terms to simplify the equation.
8x= 360
Step 2: Solve for x.
x=360
8= 45
Step 3: Find the number of male employees.
Number of male employees = 3x= 3 ×45 = 135
Therefore, there are 135 male employees in the company.
Question 19
Question
If xis directly proportional to yand inversely proportional to z, and x= 12
when y= 6 and z= 4, find xwhen y= 10 and z= 5.
11
Solution
Given that xis directly proportional to yand inversely proportional to z, we
can write:
x∝y
z
This implies that there exists a constant ksuch that:
x=ky
z
We are given that x= 12 when y= 6 and z= 4, so we can find the value of k:
12 = k6
4
k= 8
Step 1: Using the value of k, write an equation relating x,y, and z:
x= 8y
z
Step 2: Substitute y= 10 and z= 5 into the equation to find x:
x= 810
5
x= 8 ×2
x= 16
Therefore, when y= 10 and z= 5, x= 16.
Question 20
Question
A recipe calls for 3 cups of flour and 2 cups of sugar to make a certain dessert.
If you want to make 5 batches of the dessert, how many cups of sugar will you
need?
Solution
Let xrepresent the number of cups of sugar needed to make 5 batches of the
dessert.
Step 1: Set up a proportion based on the ratio of cups of flour to cups of
sugar in the recipe.
3 cups of flour
2 cups of sugar =15 cups of flour
x
12
Step 2: Cross multiply and solve for x.
3x= 2 ×15
3x= 30
x= 10
Step 3: Therefore, to make 5 batches of the dessert, you will need 10 cups
of sugar.
Question 21
Question
If 3 liters of a solution contain 15g of salt, how many grams of salt are there in
9 liters of the solution?
Solution
Step 1: Find the ratio of salt to solution in the given situation. Let xrepresent
the grams of salt in 9 liters of the solution. We know that the ratio of salt to
solution is the same in both cases, so we can set up the following proportion:
15 grams
3 liters =xgrams
9 liters
Step 2: Solve for xusing the proportion. Cross-multiplying gives:
15 ×9=3x
Step 3: Calculate the value of x.
135 = 3x
x=135
3
x= 45
Therefore, there are 45 grams of salt in 9 liters of the solution.
Question 22
Question
A group of students decided to share the cost of a 12-serving cake equally. If 5
students did not show up, each of the remaining students had to pay 1.5 times
the original amount. How many students did not show up?
13
Solution
Step 1: Let xbe the original cost per student when all students show up, and
let nbe the total number of students.
Step 2: Initially, the cost per student is x. Therefore, the total cost of the
cake is 12x.
Step 3: When 5 students did not show up, the number of students who
shared the cake became n−5. Each of the remaining students had to pay 1.5x.
Step 4: The new total cost of the cake is (n−5)(1.5x).
Step 5: Since the total cost of the cake remains the same whether all students
show up or not, we can set up an equation:
12x= (n−5)(1.5x)
Step 6: Simplifying the equation gives:
12x= 1.5nx −7.5x
Step 7: Rearranging the equation gives:
12x+ 7.5x= 1.5nx
Step 8: Combining like terms gives:
19.5x= 1.5nx
Step 9: Dividing both sides by xgives:
19.5=1.5n
Step 10: Finally, solving for ngives:
n=19.5
1.5= 13
Step 11: Therefore, the total number of students initially is 13, and the
number of students who did not show up is 5.
Question 23
Question
If 5 men can complete a construction project in 12 days, how many days will it
take for 8 men to complete the same project?
14
Solution
Step 1: Let’s denote the number of days it would take for 8 men to complete
the project as d.
Step 2: We can set up a proportion based on the number of men and the
number of days taken to complete the project:
5 men
12 days =8 men
ddays
Step 3: Cross-multiply to solve for d:
5·d= 12 ·8
Step 4: Simplify and solve for d:
5d= 96
d=96
5= 19.2
Step 5: Therefore, it will take 8 men approximately 19.2 days to complete
the construction project.
Question 24
Question
Solve for x:3
x−1=2
x+2
Solution
Step 1: Cross multiply to eliminate the denominators:
(3)(x+ 2) = (2)(x−1)
Step 2: Expand both sides:
3x+ 6 = 2x−2
Step 3: Move all terms involving xto one side:
3x−2x=−2−6
Step 4: Simplify the equation:
x=−8
15
Step 5: Check the solution by substituting x=−8 back into the original
equation: 3
(−8) −1=2
(−8) + 2
3
−9=2
−6
−1
3=−1
3
Since the left and right sides are equal, x=−8 is the correct solution.
Question 25
Question
Solve the following proportion for x:2
5=x−3
4x+1 .
Solution
We can solve the proportion by cross multiplying.
Step 1: Cross multiply to get 2(4x+ 1) = 5(x−3).
2(4x+ 1) = 5(x−3)
8x+ 2 = 5x−15
Step 2: Subtract 5xfrom both sides and add 15 to both sides.
8x+ 2 = 5x−15
8x−5x=−15 −2
3x=−17
Step 3: Divide by 3 to solve for x.
3x=−17
x=−17
3
Step 4: Simplify the result.
x=−17
3
16
Question 26
Question
Simplify the following ratio and express it in the form a:b, where aand bare
integers with no common factors other than 1:
2×53×7
32×5×112
Solution
To simplify the given ratio, we will first simplify the numerator and denominator
separately and then divide them to get the final ratio.
Step 1: Simplify the numerator
2×53×7=2×125 ×7 = 1750
Step 2: Simplify the denominator
32×5×112= 9 ×5×121 = 45 ×121 = 5445
Step 3: Divide the simplified numerator and denominator to find
the ratio
1750
5445 =350
1089 =50 ×7
33 ×33 =50 ×7
3×11 ×3×11 =50
3×11×7
3×11 =50
33×7
33 =50 ×7
33 ×33 =50
33
Question 27
Question
If two numbers are in the ratio 3 : 5 and their sum is 128, find the two numbers.
Solution
Step 1: Let the two numbers be 3xand 5x, where xis a common multiplier.
Step 2: Since their sum is 128, we can write the equation:
3x+ 5x= 128
Step 3: Simplify the equation to find the value of x:
8x= 128
Step 4: Divide both sides by 8 to solve for x:
x=128
8
17
x= 16
Step 5: Now, we can find the two numbers by substituting x= 16 back into
3xand 5x: The two numbers are:
3(16) = 48
5(16) = 80
Step 6: Therefore, the two numbers are 48 and 80.
Question 28
Question
If x:y= 3 : 4 and y:z= 5 : 2, what is the ratio of x:y:z?
Solution
To find the ratio of x:y:z, we need to combine the given ratios x:yand y:z.
Step 1: Determine a common term to connect the two ratios.
Since yis in both ratios, we can use it as the connecting term.
Step 2: Use the given ratios to rewrite x:yand y:z.
From the given ratios:
x:y= 3 : 4
y:z= 5 : 2
Step 3: Combine the two ratios to find x:y:z.
Given that x:y= 3 : 4 and y:z= 5 : 2, we can combine them to find
x:y:z:
x:y:z=3:4:2
Therefore, the ratio of x:y:zis 3 : 4 : 2 .
Question 29
Question
If 3 liters of a solution contains 15
18
Solution
Step 1: Let’s first determine how much salt is in the initial solution of 3 liters.
Step 2: Calculate the amount of salt in the initial solution. Step 3: Set up a
proportion to find the amount of water needed to dilute the solution. Step 4:
Solve the proportion to find the amount of water required.
Step 1: Determine the amount of salt in the initial solution. Let xbe the
amount of salt in 3 liters of the initial solution.
Step 2: Calculate the amount of salt in the initial solution. Since the initial
solution is 15
x= 0.15 ×3 = 0.45 liters
Now, we need to reduce the concentration of salt to 10
Step 3: Set up a proportion to find the amount of water needed to dilute
the solution. Let ybe the amount of water needed to dilute the solution.
We have the proportion: 0.45
3 + y= 0.10
Step 4: Solve the proportion to find the amount of water required. Cross
multiply to solve the proportion:
0.45 = 0.10(3 + y)
0.45 = 0.30 + 0.10y
0.10y= 0.15
y= 1.5 liters
Therefore, 1.5 liters of water must be added to the solution to reduce the
concentration of salt to 10
Question 30
Question
If a car travels 210 miles in 3 hours, how long will it take for the car to travel
560 miles?
Solution
Let xrepresent the time (in hours) it will take for the car to travel 560 miles.
We can set up a proportion to solve for x.
Step 1: Set up the proportion using the given information.
210
3=560
x
19
Step 2: Cross multiply to solve for x.
210x= 3 ×560
210x= 1680
Step 3: Solve for x.
x=1680
210
x= 8
Step 4: Final Answer: The car will take 8 hours to travel 560 miles.
Question 31
Question
Solve for xin the proportion 3
x+4 =x
6.
Solution
Step 1: Multiply both sides of the equation by 6(x+ 4) to eliminate the denom-
inators. 3
x+ 4 ·6(x+ 4) = x
6·6(x+ 4)
18 = x(x+ 4)
Step 2: Expand the right side of the equation.
18 = x2+ 4x
Step 3: Rearrange the equation into standard form.
x2+ 4x−18 = 0
Step 4: Solve the quadratic equation by factoring or using the quadratic
formula.
x2+ 6x−2x−18 = 0
x(x+ 6) −2(x+ 6) = 0
(x−2)(x+ 6) = 0
Step 5: Set each factor to zero and solve for x.
x−2 = 0 or x+ 6 = 0
x= 2 or x=−6
20
Step 6: Check the solutions to ensure they are valid in the original propor-
tion. When x= 2: 3
2+4 =2
6
3
6=2
6
1
2=1
3
When x=−6: 3
−6+4 =−6
6
3
−2=−1
This solution is extraneous.
Therefore, the solution to the proportion is x= 2.
Question 32
Question
If 4 men can build a wall in 10 days, and 6 women can build the same wall in
8 days, how many days will it take for 2 men and 3 women to build the wall
together?
Solution
Step 1: Calculate the rate at which each man and each woman can build the
wall. Let xbe the number of days for 1 man to build the wall, and ybe the
number of days for 1 woman to build the wall. Using the given information: -
4 men can build the wall in 10 days, so the rate for each man is 1
4·10 =1
40 walls
per day. - 6 women can build the wall in 8 days, so the rate for each woman is
1
6·8=1
48 walls per day.
Step 2: Set up the ratio and proportion equation. Let Dbe the number of
days for 2 men and 3 women to build the wall together. The combined rate for
2 men and 3 women working together is:
2·1
40 + 3 ·1
48 =1
20 +1
16 =9
160 walls per day
Using the formula for work, rate ×time = work, we have:
9
160 ×D= 1
9D
160 = 1
21
Step 3: Solve for the number of days, D. Multiplying both sides by 160, we
get:
9D= 160
D=160
9≈17.78
Therefore, it will take approximately 17.78 days for 2 men and 3 women to
build the wall together.
Question 33
Question
A recipe for a fruit salad calls for 2 cups of blueberries for every 3 cups of
strawberries. If we have 8 cups of blueberries, how many cups of strawberries
should we use to maintain the same ratio of blueberries to strawberries?
Solution
Step 1: Determine the ratio of blueberries to strawberries in the original recipe.
Let the number of cups of blueberries be 2xand the number of cups of straw-
berries be 3x, where xis a constant representing the ratio. Thus, the ratio of
blueberries to strawberries in the original recipe is 2x: 3xor 2 : 3.
Step 2: Calculate the number of cups of strawberries needed to maintain the
same ratio with 8 cups of blueberries. Since we have 8 cups of blueberries, set
up a proportion to find the number of cups of strawberries needed:
8 cups of blueberries
2=xcups of strawberries
3
4 = x
3
x= 12
Therefore, we need 3×12 = 36 cups of strawberries to maintain the same ratio
with 8 cups of blueberries.
Question 34
Question
Solve the following proportion for x:
3
x=x+ 4
8
22
Solution
To solve the proportion, we cross-multiply the terms to eliminate the fractions
and then solve for x.
Step 1: Cross-multiply the terms in the proportion:
3·8 = x·(x+ 4)
Step 2: Simplify both sides of the equation:
24 = x2+ 4x
Step 3: Rearrange the equation into a quadratic form:
x2+ 4x−24 = 0
Step 4: Factor the quadratic equation:
(x+ 6)(x−4) = 0
Step 5: Set each factor to zero and solve for x:
x+ 6 = 0 or x−4 = 0
Step 6: Solve for x: If x+ 6 = 0, then x=−6. If x−4 = 0, then x= 4.
Step 7: Check the solutions by substituting back into the original equation:
For x=−6: 3
−6=−6+4
8
1
−2=−2
8
−1
2=−1
4
This solution is not valid.
For x= 4: 3
4=4+4
8
3
4=8
8
3
4= 1
This solution is valid.
Therefore, the solution to the proportion is x= 4.
Question 35
Question
Solve the following proportion for x:3
2x+4 =x+1
5
23
Solution
To solve the proportion, we will cross multiply and simplify the resulting equa-
tion.
Step 1: Cross multiply to get rid of the fractions.
(3)(5) = (2x+ 4)(x+ 1)
Step 2: Expand both sides of the equation.
15 = 2x2+ 2x+ 4x+ 4
Step 3: Combine like terms on the right side.
15 = 2x2+ 6x+ 4
Step 4: Rearrange the equation to set it equal to zero.
2x2+ 6x+ 4 −15 = 0
2x2+ 6x−11 = 0
Step 5: Use the quadratic formula to solve for x.
x=−6±p62−4(2)(−11)
2(2)
x=−6±√36 + 88
4
x=−6±√124
4
x=−6±2√31
4
x=−3±√31
2
So, the solutions for xare x=−3+√31
2and x=−3−√31
2.
24
Solution
Step 1: Simplify the numerator and denominator separately. Step 2: Divide the
numerator by the denominator.
Step 1: To simplify the numerator 4x2y, we rewrite it as 4 ·x·x·y.
To simplify the denominator 12xy3, we rewrite it as 12 ·x·y·y·y.
Step 2: Dividing the numerator by the denominator:
4x2y
12xy3=4·x·x·y
12 ·x·y·y·y=1
3·x
y2=x
3y2
Question 3
Question
Solve the following proportion for x:
2x+ 1
3x−2=3x+ 5
4x−1
Solution
To solve the given proportion for x, we will first cross multiply and then simplify
the resulting equation step by step.
Step 1: Cross multiply to eliminate the denominators:
(2x+ 1)(4x−1) = (3x−2)(3x+ 5)
Step 2: Expand both sides of the equation:
8x2−2x+ 4x−1=9x2+ 15x−6x−10
Step 3: Simplify both sides of the equation:
8x2+ 2x−1=9x2+ 9x−10
Step 4: Rearrange the equation to set it equal to zero:
0 = x2+ 7x−9
Step 5: Solve the quadratic equation by factoring or using the quadratic
formula:
0=(x+ 9)(x−1)
Step 6: Set each factor to zero and solve for x:
x+ 9 = 0 =⇒x=−9 or x−1 = 0 =⇒x= 1
2
Step 7: Check for extraneous solutions by substituting each back into the
original proportion: Checking x=−9:
2(−9) + 1
3(−9) −2=3(−9) + 5
4(−9) −1
−17
−29 =−22
−37
17
29 =22
37
Checking x= 1:
2(1) + 1
3(1) −2=3(1) + 5
4(1) −1
3
1=8
3
3 = 22
3
Therefore, the solution to the proportion is x= 1.
Question 4
Question
Solve for xgiven that 3x+1
5x−2=5
9.
Solution
Step 1: Cross multiply to get rid of the fractions.
3x+ 1
5x−2=5
9
9(3x+ 1) = 5(5x−2)
Step 2: Expand both sides of the equation.
27x+ 9 = 25x−10
Step 3: Simplify the equation by collecting like terms.
27x+ 9 = 25x−10
27x−25x=−10 −9
2x=−19
Step 4: Solve for xby dividing both sides by 2.
2x=−19
x=−19
2
Therefore, the solution is x=−19
2.
3
Question 5
Question
A construction crew is building a scale model of a house. If 2.5 feet of a wall of
the actual house corresponds to 0.1 inches on the model, how many inches on
the model represent 9 feet of the actual wall?
Solution
Step 1: Calculate the scale factor. Let xbe the number of inches on the model
that represent 9 feet of the actual wall. We can set up a proportion to find the
scale factor: 2.5 feet
0.1 inches =9 feet
xinches
Step 2: Solve for x. Cross multiply to solve for x:
2.5×x= 0.1×9
2.5x= 0.9
Step 3: Calculate the value of x. Divide by 2.5 to find the value of x:
x=0.9
2.5
x= 0.36
Therefore, 0.36 inches on the model represent 9 feet of the actual wall.
Question 6
Question
Solve the following proportion for x:
2x+ 3
5=4x+ 7
9
Solution
Step 1: Cross multiply to eliminate the fractions:
9(2x+ 3) = 5(4x+ 7)
18x+ 27 = 20x+ 35
Step 2: Rearrange the equation by isolating xterms on one side:
18x−20x= 35 −27
−2x= 8
4
Step 3: Divide by the coefficient of xto solve for x:
x=8
−2
x=−4
Therefore, the solution to the proportion is x=−4.
Question 7
Question
A recipe requires 3 cups of flour for every 2 cups of sugar. If you have 8 cups of
flour, how many cups of sugar do you need?
Solution
Step 1: Calculate the ratio of flour to sugar in the recipe. Let xrepresent the
number of cups of sugar needed. The ratio of flour to sugar in the recipe is 3
2.
This means that 8
x=3
2.
Step 2: Solve for x. Cross multiply to solve for x: 2 ∗8=3x
16 = 3x
x=16
3
x= 51
3
Step 3: Interpret the solution. You need 16
3or 51
3cups of sugar to go with
8 cups of flour in the recipe.
Question 8
Question
Solve the proportion: 2x+1
3=4x−5
5.
Solution
To solve the proportion 2x+1
3=4x−5
5, we will cross-multiply to eliminate the
fractions and then solve for x.
Step 1: Cross-multiply to get: (2x+ 1) ×5=3×(4x−5).
Step 2: Expand both sides: 10x+ 5 = 12x−15.
Step 3: Rearrange the equation to isolate xterms: 10x+ 5 = 12x−15 ⇒
10x+ 15 = 12x−5.
Step 4: Move all terms involving xto one side: 10x−12x=−5−15 ⇒
−2x=−20.
Step 5: Solve for x:−2x=−20 ⇒x=−20
−2⇒x= 10.
Step 6: Check the solution by substituting x= 10 back into the original
proportion.
5
2(10)+1
3=4(10)−5
5
20+1
3=40−5
5
21
3=35
5
7 = 7.
Since the equation holds true, x= 10 is the correct solution.
Question 9
Question
Solve the following proportion for x:
3
x+ 2 =x+ 1
5
Solution
Step 1: Cross multiply to get rid of the fractions.
3·5=(x+ 1)(x+ 2)
Step 2: Simplify the equation.
15 = x2+ 3x+ 2
Step 3: Rearrange the equation into standard quadratic form.
x2+ 3x−13 = 0
Step 4: Use the quadratic formula to solve for x:
x=−b±√b2−4ac
2a
Step 5: Plug in a= 1, b= 3, and c=−13 into the quadratic formula.
x=−3±p32−4(1)(−13)
2(1)
Step 6: Simplify under the square root.
x=−3±√9 + 52
2
x=−3±√61
2
Therefore, the solutions for xare
x=−3 + √61
2and x=−3−√61
2
6
Question 10
Question
Simplify the following expression to its simplest form: 3x2+ 2xy
5x2−3xy .
Solution
Step 1: Factor out xfrom both the numerator and denominator:
x(3x+ 2y)
x(5x−3y)
Step 2: Simplify by canceling out the common factors.
3x+ 2y
5x−3y
Therefore, the simplified form of the expression is 3x+ 2y
5x−3y.
Question 11
Question
Given that xvaries directly with yand inversely with z, and that x= 6 when
y= 3 and z= 4, find xwhen y= 10 and z= 6.
Solution
Step 1: First, we express the direct and inverse variation relationships using the
proportionality constants: Let k1be the constant of direct variation and k2be
the constant of inverse variation. Then, we have the relationships:
x=k1y
x=k2
z
Step 2: Next, we use the given information to find the values of k1and k2:
When x= 6, y= 3, and z= 4, we have:
6 = k1·3
6 = k2
4
Solving these equations gives k1= 2 and k2= 24.
Step 3: Now that we have the proportionality constants, we can find the
value of xwhen y= 10 and z=6:x= 2 ·10 = 20
x=24
6= 4
Therefore, when y= 10 and z= 6, x= 20 or x= 4.
7
Question 12
Question
Solve the following proportion for x:
2x+ 4
3=x+ 5
6
Solution
Step 1: Cross multiply to eliminate the fractions:
6(2x+ 4) = 3(x+ 5)
12x+ 24 = 3x+ 15
Step 2: Simplify the equation:
12x−3x= 15 −24
9x=−9
Step 3: Solve for x:
x=−9
9
x=−1
Thus, the solution to the proportion is x=−1.
Question 13
Question
Samantha and Emily decide to split a sum of money in the ratio 5:3. If Samantha
receives 120morethanEmily, howmuchmoneydideachreceive?
Solution
Step 1: Let’s denote the amount of money that Samantha and Emily re-
ceived as 5xand 3x, respectively. We are also given that Samantha received
120morethanEmily, sowecansetuptheequation : 5x= 3x+ 120
Step 2: Now, we can solve for x:
5x= 3x+ 120
2x= 120
x= 60
8
Step 3: Now that we have found x, we can find out how much money each
person received:
Samantha received 5x= 5(60) = $300
Emily received 3x= 3(60) = $180
Therefore, Samantha received
$
300 and Emily received
$
180.
Question 14
Question
If 3 liters of a solution contains 20
Solution
Step 1: Determine the amount of salt in the initial solution.
Let xrepresent the amount of salt in the 3 liters of 20
0.20 ×3 = x
x= 0.6 liters
Step 2: Set up the equation for the final solution.
Let ybe the amount of pure salt added to the solution. The total volume of
the final solution is 3 + yliters. We want the final solution to be 25
x+y
3 + y= 0.25
Substitute the value of xwe found in Step 1:
0.6 + y
3 + y= 0.25
Step 3: Solve for y.
Simplify the equation:
0.6 + y= 0.25(3 + y)
0.6 + y= 0.75 + 0.25y
0.75 −0.6=0.25y−y
0.15 = −0.75y
y=−0.15
0.75 =−0.2
Step 4: Interpret the solution.
Since the volume of salt cannot be negative, we discard the negative solution.
Therefore, we do not need to add any salt to make the solution 25
9
Question 15
Question
A recipe for bread requires 3 cups of flour, 1 cup of sugar, and 1
2cup of oil.
If you want to make 2 loaves of bread, how much flour, sugar, and oil do you
need?
Solution
Let’s first determine the ratio of ingredients needed to make one loaf of bread:
- Flour: 3 cups - Sugar: 1 cup - Oil: 1
2cup
To make 2 loaves of bread, we simply double the amount of each ingredient:
- Flour: 3 cups×2 = 6 cups - Sugar: 1 cup×2 = 2 cups - Oil: 1
2cup×2 = 1 cup
Therefore, to make 2 loaves of bread, you will need 6 cups of flour, 2 cups
of sugar, and 1 cup of oil.
Question 16
Question
If x,y, and zare positive numbers such that x
y=3
4and y
z=5
6, find the ratio
x
z.
Solution
Step 1: From x
y=3
4, we can rewrite it as x=3
4y.
Step 2: From y
z=5
6, we can rewrite it as y=5
6z.
Step 3: Substituting the expression for yfrom Step 2 into the expression for
xfrom Step 1, we get:
x=3
45
6z=5
8z
Step 4: Therefore, the ratio x
zis:
x
z=
5
8z
z=5
8= 5 : 8
Hence, the ratio x
zis 5 : 8.
Question 17
Question
If xand yare positive real numbers such that x:y= 4 : 3 and x+5
y−2=7
5, find
the value of x.
10
Solution
Step 1: We start by setting up the proportion based on the given ratio x:y=4:
3. Step 2: This means x
y=4
3. We can rewrite this as y=3
4x. Step 3: Substitute
y=3
4xinto the second equation x+5
y−2=7
5. Step 4: We get x+5
(3
4x)−2=7
5. Step
5: Simplifying the denominator, we have x+5
3x
4−2=7
5. Step 6: To eliminate the
fractions, we can multiply both sides by 20 (the least common multiple of 4 and
5). Step 7: This gives 20(x+ 5) = 4(3x)−8. Step 8: Simplifying the equation
gives 20x+ 100 = 12x−8. Step 9: Rearranging terms, we have 8x=−108.
Step 10: Finally, solving for xgives x=−108
8=−13.5. Therefore, the value of
xis −13.5.
Question 18
Question
In a certain company, the ratio of the number of male employees to the number
of female employees is 3:5. If there are 360 employees in total, how many of
them are male?
Solution
Let the number of male employees be 3xand the number of female employees
be 5x. We are given that the total number of employees is 360. Therefore, we
have the equation:
3x+ 5x= 360
Step 1: Combine like terms to simplify the equation.
8x= 360
Step 2: Solve for x.
x=360
8= 45
Step 3: Find the number of male employees.
Number of male employees = 3x= 3 ×45 = 135
Therefore, there are 135 male employees in the company.
Question 19
Question
If xis directly proportional to yand inversely proportional to z, and x= 12
when y= 6 and z= 4, find xwhen y= 10 and z= 5.
11
Solution
Given that xis directly proportional to yand inversely proportional to z, we
can write:
x∝y
z
This implies that there exists a constant ksuch that:
x=ky
z
We are given that x= 12 when y= 6 and z= 4, so we can find the value of k:
12 = k6
4
k= 8
Step 1: Using the value of k, write an equation relating x,y, and z:
x= 8y
z
Step 2: Substitute y= 10 and z= 5 into the equation to find x:
x= 810
5
x= 8 ×2
x= 16
Therefore, when y= 10 and z= 5, x= 16.
Question 20
Question
A recipe calls for 3 cups of flour and 2 cups of sugar to make a certain dessert.
If you want to make 5 batches of the dessert, how many cups of sugar will you
need?
Solution
Let xrepresent the number of cups of sugar needed to make 5 batches of the
dessert.
Step 1: Set up a proportion based on the ratio of cups of flour to cups of
sugar in the recipe.
3 cups of flour
2 cups of sugar =15 cups of flour
x
12
Step 2: Cross multiply and solve for x.
3x= 2 ×15
3x= 30
x= 10
Step 3: Therefore, to make 5 batches of the dessert, you will need 10 cups
of sugar.
Question 21
Question
If 3 liters of a solution contain 15g of salt, how many grams of salt are there in
9 liters of the solution?
Solution
Step 1: Find the ratio of salt to solution in the given situation. Let xrepresent
the grams of salt in 9 liters of the solution. We know that the ratio of salt to
solution is the same in both cases, so we can set up the following proportion:
15 grams
3 liters =xgrams
9 liters
Step 2: Solve for xusing the proportion. Cross-multiplying gives:
15 ×9=3x
Step 3: Calculate the value of x.
135 = 3x
x=135
3
x= 45
Therefore, there are 45 grams of salt in 9 liters of the solution.
Question 22
Question
A group of students decided to share the cost of a 12-serving cake equally. If 5
students did not show up, each of the remaining students had to pay 1.5 times
the original amount. How many students did not show up?
13
Solution
Step 1: Let xbe the original cost per student when all students show up, and
let nbe the total number of students.
Step 2: Initially, the cost per student is x. Therefore, the total cost of the
cake is 12x.
Step 3: When 5 students did not show up, the number of students who
shared the cake became n−5. Each of the remaining students had to pay 1.5x.
Step 4: The new total cost of the cake is (n−5)(1.5x).
Step 5: Since the total cost of the cake remains the same whether all students
show up or not, we can set up an equation:
12x= (n−5)(1.5x)
Step 6: Simplifying the equation gives:
12x= 1.5nx −7.5x
Step 7: Rearranging the equation gives:
12x+ 7.5x= 1.5nx
Step 8: Combining like terms gives:
19.5x= 1.5nx
Step 9: Dividing both sides by xgives:
19.5=1.5n
Step 10: Finally, solving for ngives:
n=19.5
1.5= 13
Step 11: Therefore, the total number of students initially is 13, and the
number of students who did not show up is 5.
Question 23
Question
If 5 men can complete a construction project in 12 days, how many days will it
take for 8 men to complete the same project?
14
Solution
Step 1: Let’s denote the number of days it would take for 8 men to complete
the project as d.
Step 2: We can set up a proportion based on the number of men and the
number of days taken to complete the project:
5 men
12 days =8 men
ddays
Step 3: Cross-multiply to solve for d:
5·d= 12 ·8
Step 4: Simplify and solve for d:
5d= 96
d=96
5= 19.2
Step 5: Therefore, it will take 8 men approximately 19.2 days to complete
the construction project.
Question 24
Question
Solve for x:3
x−1=2
x+2
Solution
Step 1: Cross multiply to eliminate the denominators:
(3)(x+ 2) = (2)(x−1)
Step 2: Expand both sides:
3x+ 6 = 2x−2
Step 3: Move all terms involving xto one side:
3x−2x=−2−6
Step 4: Simplify the equation:
x=−8
15
Step 5: Check the solution by substituting x=−8 back into the original
equation: 3
(−8) −1=2
(−8) + 2
3
−9=2
−6
−1
3=−1
3
Since the left and right sides are equal, x=−8 is the correct solution.
Question 25
Question
Solve the following proportion for x:2
5=x−3
4x+1 .
Solution
We can solve the proportion by cross multiplying.
Step 1: Cross multiply to get 2(4x+ 1) = 5(x−3).
2(4x+ 1) = 5(x−3)
8x+ 2 = 5x−15
Step 2: Subtract 5xfrom both sides and add 15 to both sides.
8x+ 2 = 5x−15
8x−5x=−15 −2
3x=−17
Step 3: Divide by 3 to solve for x.
3x=−17
x=−17
3
Step 4: Simplify the result.
x=−17
3
16
Question 26
Question
Simplify the following ratio and express it in the form a:b, where aand bare
integers with no common factors other than 1:
2×53×7
32×5×112
Solution
To simplify the given ratio, we will first simplify the numerator and denominator
separately and then divide them to get the final ratio.
Step 1: Simplify the numerator
2×53×7=2×125 ×7 = 1750
Step 2: Simplify the denominator
32×5×112= 9 ×5×121 = 45 ×121 = 5445
Step 3: Divide the simplified numerator and denominator to find
the ratio
1750
5445 =350
1089 =50 ×7
33 ×33 =50 ×7
3×11 ×3×11 =50
3×11×7
3×11 =50
33×7
33 =50 ×7
33 ×33 =50
33
Question 27
Question
If two numbers are in the ratio 3 : 5 and their sum is 128, find the two numbers.
Solution
Step 1: Let the two numbers be 3xand 5x, where xis a common multiplier.
Step 2: Since their sum is 128, we can write the equation:
3x+ 5x= 128
Step 3: Simplify the equation to find the value of x:
8x= 128
Step 4: Divide both sides by 8 to solve for x:
x=128
8
17
x= 16
Step 5: Now, we can find the two numbers by substituting x= 16 back into
3xand 5x: The two numbers are:
3(16) = 48
5(16) = 80
Step 6: Therefore, the two numbers are 48 and 80.
Question 28
Question
If x:y= 3 : 4 and y:z= 5 : 2, what is the ratio of x:y:z?
Solution
To find the ratio of x:y:z, we need to combine the given ratios x:yand y:z.
Step 1: Determine a common term to connect the two ratios.
Since yis in both ratios, we can use it as the connecting term.
Step 2: Use the given ratios to rewrite x:yand y:z.
From the given ratios:
x:y= 3 : 4
y:z= 5 : 2
Step 3: Combine the two ratios to find x:y:z.
Given that x:y= 3 : 4 and y:z= 5 : 2, we can combine them to find
x:y:z:
x:y:z=3:4:2
Therefore, the ratio of x:y:zis 3 : 4 : 2 .
Question 29
Question
If 3 liters of a solution contains 15
18
Solution
Step 1: Let’s first determine how much salt is in the initial solution of 3 liters.
Step 2: Calculate the amount of salt in the initial solution. Step 3: Set up a
proportion to find the amount of water needed to dilute the solution. Step 4:
Solve the proportion to find the amount of water required.
Step 1: Determine the amount of salt in the initial solution. Let xbe the
amount of salt in 3 liters of the initial solution.
Step 2: Calculate the amount of salt in the initial solution. Since the initial
solution is 15
x= 0.15 ×3 = 0.45 liters
Now, we need to reduce the concentration of salt to 10
Step 3: Set up a proportion to find the amount of water needed to dilute
the solution. Let ybe the amount of water needed to dilute the solution.
We have the proportion: 0.45
3 + y= 0.10
Step 4: Solve the proportion to find the amount of water required. Cross
multiply to solve the proportion:
0.45 = 0.10(3 + y)
0.45 = 0.30 + 0.10y
0.10y= 0.15
y= 1.5 liters
Therefore, 1.5 liters of water must be added to the solution to reduce the
concentration of salt to 10
Question 30
Question
If a car travels 210 miles in 3 hours, how long will it take for the car to travel
560 miles?
Solution
Let xrepresent the time (in hours) it will take for the car to travel 560 miles.
We can set up a proportion to solve for x.
Step 1: Set up the proportion using the given information.
210
3=560
x
19
Step 2: Cross multiply to solve for x.
210x= 3 ×560
210x= 1680
Step 3: Solve for x.
x=1680
210
x= 8
Step 4: Final Answer: The car will take 8 hours to travel 560 miles.
Question 31
Question
Solve for xin the proportion 3
x+4 =x
6.
Solution
Step 1: Multiply both sides of the equation by 6(x+ 4) to eliminate the denom-
inators. 3
x+ 4 ·6(x+ 4) = x
6·6(x+ 4)
18 = x(x+ 4)
Step 2: Expand the right side of the equation.
18 = x2+ 4x
Step 3: Rearrange the equation into standard form.
x2+ 4x−18 = 0
Step 4: Solve the quadratic equation by factoring or using the quadratic
formula.
x2+ 6x−2x−18 = 0
x(x+ 6) −2(x+ 6) = 0
(x−2)(x+ 6) = 0
Step 5: Set each factor to zero and solve for x.
x−2 = 0 or x+ 6 = 0
x= 2 or x=−6
20
Step 6: Check the solutions to ensure they are valid in the original propor-
tion. When x= 2: 3
2+4 =2
6
3
6=2
6
1
2=1
3
When x=−6: 3
−6+4 =−6
6
3
−2=−1
This solution is extraneous.
Therefore, the solution to the proportion is x= 2.
Question 32
Question
If 4 men can build a wall in 10 days, and 6 women can build the same wall in
8 days, how many days will it take for 2 men and 3 women to build the wall
together?
Solution
Step 1: Calculate the rate at which each man and each woman can build the
wall. Let xbe the number of days for 1 man to build the wall, and ybe the
number of days for 1 woman to build the wall. Using the given information: -
4 men can build the wall in 10 days, so the rate for each man is 1
4·10 =1
40 walls
per day. - 6 women can build the wall in 8 days, so the rate for each woman is
1
6·8=1
48 walls per day.
Step 2: Set up the ratio and proportion equation. Let Dbe the number of
days for 2 men and 3 women to build the wall together. The combined rate for
2 men and 3 women working together is:
2·1
40 + 3 ·1
48 =1
20 +1
16 =9
160 walls per day
Using the formula for work, rate ×time = work, we have:
9
160 ×D= 1
9D
160 = 1
21
Step 3: Solve for the number of days, D. Multiplying both sides by 160, we
get:
9D= 160
D=160
9≈17.78
Therefore, it will take approximately 17.78 days for 2 men and 3 women to
build the wall together.
Question 33
Question
A recipe for a fruit salad calls for 2 cups of blueberries for every 3 cups of
strawberries. If we have 8 cups of blueberries, how many cups of strawberries
should we use to maintain the same ratio of blueberries to strawberries?
Solution
Step 1: Determine the ratio of blueberries to strawberries in the original recipe.
Let the number of cups of blueberries be 2xand the number of cups of straw-
berries be 3x, where xis a constant representing the ratio. Thus, the ratio of
blueberries to strawberries in the original recipe is 2x: 3xor 2 : 3.
Step 2: Calculate the number of cups of strawberries needed to maintain the
same ratio with 8 cups of blueberries. Since we have 8 cups of blueberries, set
up a proportion to find the number of cups of strawberries needed:
8 cups of blueberries
2=xcups of strawberries
3
4 = x
3
x= 12
Therefore, we need 3×12 = 36 cups of strawberries to maintain the same ratio
with 8 cups of blueberries.
Question 34
Question
Solve the following proportion for x:
3
x=x+ 4
8
22
Solution
To solve the proportion, we cross-multiply the terms to eliminate the fractions
and then solve for x.
Step 1: Cross-multiply the terms in the proportion:
3·8 = x·(x+ 4)
Step 2: Simplify both sides of the equation:
24 = x2+ 4x
Step 3: Rearrange the equation into a quadratic form:
x2+ 4x−24 = 0
Step 4: Factor the quadratic equation:
(x+ 6)(x−4) = 0
Step 5: Set each factor to zero and solve for x:
x+ 6 = 0 or x−4 = 0
Step 6: Solve for x: If x+ 6 = 0, then x=−6. If x−4 = 0, then x= 4.
Step 7: Check the solutions by substituting back into the original equation:
For x=−6: 3
−6=−6+4
8
1
−2=−2
8
−1
2=−1
4
This solution is not valid.
For x= 4: 3
4=4+4
8
3
4=8
8
3
4= 1
This solution is valid.
Therefore, the solution to the proportion is x= 4.
Question 35
Question
Solve the following proportion for x:3
2x+4 =x+1
5
23
Solution
To solve the proportion, we will cross multiply and simplify the resulting equa-
tion.
Step 1: Cross multiply to get rid of the fractions.
(3)(5) = (2x+ 4)(x+ 1)
Step 2: Expand both sides of the equation.
15 = 2x2+ 2x+ 4x+ 4
Step 3: Combine like terms on the right side.
15 = 2x2+ 6x+ 4
Step 4: Rearrange the equation to set it equal to zero.
2x2+ 6x+ 4 −15 = 0
2x2+ 6x−11 = 0
Step 5: Use the quadratic formula to solve for x.
x=−6±p62−4(2)(−11)
2(2)
x=−6±√36 + 88
4
x=−6±√124
4
x=−6±2√31
4
x=−3±√31
2
So, the solutions for xare x=−3+√31
2and x=−3−√31
2.
24
Solution
Step 1: Simplify the numerator and denominator separately. Step 2: Divide the
numerator by the denominator.
Step 1: To simplify the numerator 4x2y, we rewrite it as 4 ·x·x·y.
To simplify the denominator 12xy3, we rewrite it as 12 ·x·y·y·y.
Step 2: Dividing the numerator by the denominator:
4x2y
12xy3=4·x·x·y
12 ·x·y·y·y=1
3·x
y2=x
3y2
Question 3
Question
Solve the following proportion for x:
2x+ 1
3x−2=3x+ 5
4x−1
Solution
To solve the given proportion for x, we will first cross multiply and then simplify
the resulting equation step by step.
Step 1: Cross multiply to eliminate the denominators:
(2x+ 1)(4x−1) = (3x−2)(3x+ 5)
Step 2: Expand both sides of the equation:
8x2−2x+ 4x−1=9x2+ 15x−6x−10
Step 3: Simplify both sides of the equation:
8x2+ 2x−1=9x2+ 9x−10
Step 4: Rearrange the equation to set it equal to zero:
0 = x2+ 7x−9
Step 5: Solve the quadratic equation by factoring or using the quadratic
formula:
0=(x+ 9)(x−1)
Step 6: Set each factor to zero and solve for x:
x+ 9 = 0 =⇒x=−9 or x−1 = 0 =⇒x= 1
2
Step 7: Check for extraneous solutions by substituting each back into the
original proportion: Checking x=−9:
2(−9) + 1
3(−9) −2=3(−9) + 5
4(−9) −1
−17
−29 =−22
−37
17
29 =22
37
Checking x= 1:
2(1) + 1
3(1) −2=3(1) + 5
4(1) −1
3
1=8
3
3 = 22
3
Therefore, the solution to the proportion is x= 1.
Question 4
Question
Solve for xgiven that 3x+1
5x−2=5
9.
Solution
Step 1: Cross multiply to get rid of the fractions.
3x+ 1
5x−2=5
9
9(3x+ 1) = 5(5x−2)
Step 2: Expand both sides of the equation.
27x+ 9 = 25x−10
Step 3: Simplify the equation by collecting like terms.
27x+ 9 = 25x−10
27x−25x=−10 −9
2x=−19
Step 4: Solve for xby dividing both sides by 2.
2x=−19
x=−19
2
Therefore, the solution is x=−19
2.
3
Question 5
Question
A construction crew is building a scale model of a house. If 2.5 feet of a wall of
the actual house corresponds to 0.1 inches on the model, how many inches on
the model represent 9 feet of the actual wall?
Solution
Step 1: Calculate the scale factor. Let xbe the number of inches on the model
that represent 9 feet of the actual wall. We can set up a proportion to find the
scale factor: 2.5 feet
0.1 inches =9 feet
xinches
Step 2: Solve for x. Cross multiply to solve for x:
2.5×x= 0.1×9
2.5x= 0.9
Step 3: Calculate the value of x. Divide by 2.5 to find the value of x:
x=0.9
2.5
x= 0.36
Therefore, 0.36 inches on the model represent 9 feet of the actual wall.
Question 6
Question
Solve the following proportion for x:
2x+ 3
5=4x+ 7
9
Solution
Step 1: Cross multiply to eliminate the fractions:
9(2x+ 3) = 5(4x+ 7)
18x+ 27 = 20x+ 35
Step 2: Rearrange the equation by isolating xterms on one side:
18x−20x= 35 −27
−2x= 8
4
Step 3: Divide by the coefficient of xto solve for x:
x=8
−2
x=−4
Therefore, the solution to the proportion is x=−4.
Question 7
Question
A recipe requires 3 cups of flour for every 2 cups of sugar. If you have 8 cups of
flour, how many cups of sugar do you need?
Solution
Step 1: Calculate the ratio of flour to sugar in the recipe. Let xrepresent the
number of cups of sugar needed. The ratio of flour to sugar in the recipe is 3
2.
This means that 8
x=3
2.
Step 2: Solve for x. Cross multiply to solve for x: 2 ∗8=3x
16 = 3x
x=16
3
x= 51
3
Step 3: Interpret the solution. You need 16
3or 51
3cups of sugar to go with
8 cups of flour in the recipe.
Question 8
Question
Solve the proportion: 2x+1
3=4x−5
5.
Solution
To solve the proportion 2x+1
3=4x−5
5, we will cross-multiply to eliminate the
fractions and then solve for x.
Step 1: Cross-multiply to get: (2x+ 1) ×5=3×(4x−5).
Step 2: Expand both sides: 10x+ 5 = 12x−15.
Step 3: Rearrange the equation to isolate xterms: 10x+ 5 = 12x−15 ⇒
10x+ 15 = 12x−5.
Step 4: Move all terms involving xto one side: 10x−12x=−5−15 ⇒
−2x=−20.
Step 5: Solve for x:−2x=−20 ⇒x=−20
−2⇒x= 10.
Step 6: Check the solution by substituting x= 10 back into the original
proportion.
5
2(10)+1
3=4(10)−5
5
20+1
3=40−5
5
21
3=35
5
7 = 7.
Since the equation holds true, x= 10 is the correct solution.
Question 9
Question
Solve the following proportion for x:
3
x+ 2 =x+ 1
5
Solution
Step 1: Cross multiply to get rid of the fractions.
3·5=(x+ 1)(x+ 2)
Step 2: Simplify the equation.
15 = x2+ 3x+ 2
Step 3: Rearrange the equation into standard quadratic form.
x2+ 3x−13 = 0
Step 4: Use the quadratic formula to solve for x:
x=−b±√b2−4ac
2a
Step 5: Plug in a= 1, b= 3, and c=−13 into the quadratic formula.
x=−3±p32−4(1)(−13)
2(1)
Step 6: Simplify under the square root.
x=−3±√9 + 52
2
x=−3±√61
2
Therefore, the solutions for xare
x=−3 + √61
2and x=−3−√61
2
6
Question 10
Question
Simplify the following expression to its simplest form: 3x2+ 2xy
5x2−3xy .
Solution
Step 1: Factor out xfrom both the numerator and denominator:
x(3x+ 2y)
x(5x−3y)
Step 2: Simplify by canceling out the common factors.
3x+ 2y
5x−3y
Therefore, the simplified form of the expression is 3x+ 2y
5x−3y.
Question 11
Question
Given that xvaries directly with yand inversely with z, and that x= 6 when
y= 3 and z= 4, find xwhen y= 10 and z= 6.
Solution
Step 1: First, we express the direct and inverse variation relationships using the
proportionality constants: Let k1be the constant of direct variation and k2be
the constant of inverse variation. Then, we have the relationships:
x=k1y
x=k2
z
Step 2: Next, we use the given information to find the values of k1and k2:
When x= 6, y= 3, and z= 4, we have:
6 = k1·3
6 = k2
4
Solving these equations gives k1= 2 and k2= 24.
Step 3: Now that we have the proportionality constants, we can find the
value of xwhen y= 10 and z=6:x= 2 ·10 = 20
x=24
6= 4
Therefore, when y= 10 and z= 6, x= 20 or x= 4.
7
Question 12
Question
Solve the following proportion for x:
2x+ 4
3=x+ 5
6
Solution
Step 1: Cross multiply to eliminate the fractions:
6(2x+ 4) = 3(x+ 5)
12x+ 24 = 3x+ 15
Step 2: Simplify the equation:
12x−3x= 15 −24
9x=−9
Step 3: Solve for x:
x=−9
9
x=−1
Thus, the solution to the proportion is x=−1.
Question 13
Question
Samantha and Emily decide to split a sum of money in the ratio 5:3. If Samantha
receives 120morethanEmily, howmuchmoneydideachreceive?
Solution
Step 1: Let’s denote the amount of money that Samantha and Emily re-
ceived as 5xand 3x, respectively. We are also given that Samantha received
120morethanEmily, sowecansetuptheequation : 5x= 3x+ 120
Step 2: Now, we can solve for x:
5x= 3x+ 120
2x= 120
x= 60
8
Step 3: Now that we have found x, we can find out how much money each
person received:
Samantha received 5x= 5(60) = $300
Emily received 3x= 3(60) = $180
Therefore, Samantha received
$
300 and Emily received
$
180.
Question 14
Question
If 3 liters of a solution contains 20
Solution
Step 1: Determine the amount of salt in the initial solution.
Let xrepresent the amount of salt in the 3 liters of 20
0.20 ×3 = x
x= 0.6 liters
Step 2: Set up the equation for the final solution.
Let ybe the amount of pure salt added to the solution. The total volume of
the final solution is 3 + yliters. We want the final solution to be 25
x+y
3 + y= 0.25
Substitute the value of xwe found in Step 1:
0.6 + y
3 + y= 0.25
Step 3: Solve for y.
Simplify the equation:
0.6 + y= 0.25(3 + y)
0.6 + y= 0.75 + 0.25y
0.75 −0.6=0.25y−y
0.15 = −0.75y
y=−0.15
0.75 =−0.2
Step 4: Interpret the solution.
Since the volume of salt cannot be negative, we discard the negative solution.
Therefore, we do not need to add any salt to make the solution 25
9
Question 15
Question
A recipe for bread requires 3 cups of flour, 1 cup of sugar, and 1
2cup of oil.
If you want to make 2 loaves of bread, how much flour, sugar, and oil do you
need?
Solution
Let’s first determine the ratio of ingredients needed to make one loaf of bread:
- Flour: 3 cups - Sugar: 1 cup - Oil: 1
2cup
To make 2 loaves of bread, we simply double the amount of each ingredient:
- Flour: 3 cups×2 = 6 cups - Sugar: 1 cup×2 = 2 cups - Oil: 1
2cup×2 = 1 cup
Therefore, to make 2 loaves of bread, you will need 6 cups of flour, 2 cups
of sugar, and 1 cup of oil.
Question 16
Question
If x,y, and zare positive numbers such that x
y=3
4and y
z=5
6, find the ratio
x
z.
Solution
Step 1: From x
y=3
4, we can rewrite it as x=3
4y.
Step 2: From y
z=5
6, we can rewrite it as y=5
6z.
Step 3: Substituting the expression for yfrom Step 2 into the expression for
xfrom Step 1, we get:
x=3
45
6z=5
8z
Step 4: Therefore, the ratio x
zis:
x
z=
5
8z
z=5
8= 5 : 8
Hence, the ratio x
zis 5 : 8.
Question 17
Question
If xand yare positive real numbers such that x:y= 4 : 3 and x+5
y−2=7
5, find
the value of x.
10
Solution
Step 1: We start by setting up the proportion based on the given ratio x:y=4:
3. Step 2: This means x
y=4
3. We can rewrite this as y=3
4x. Step 3: Substitute
y=3
4xinto the second equation x+5
y−2=7
5. Step 4: We get x+5
(3
4x)−2=7
5. Step
5: Simplifying the denominator, we have x+5
3x
4−2=7
5. Step 6: To eliminate the
fractions, we can multiply both sides by 20 (the least common multiple of 4 and
5). Step 7: This gives 20(x+ 5) = 4(3x)−8. Step 8: Simplifying the equation
gives 20x+ 100 = 12x−8. Step 9: Rearranging terms, we have 8x=−108.
Step 10: Finally, solving for xgives x=−108
8=−13.5. Therefore, the value of
xis −13.5.
Question 18
Question
In a certain company, the ratio of the number of male employees to the number
of female employees is 3:5. If there are 360 employees in total, how many of
them are male?
Solution
Let the number of male employees be 3xand the number of female employees
be 5x. We are given that the total number of employees is 360. Therefore, we
have the equation:
3x+ 5x= 360
Step 1: Combine like terms to simplify the equation.
8x= 360
Step 2: Solve for x.
x=360
8= 45
Step 3: Find the number of male employees.
Number of male employees = 3x= 3 ×45 = 135
Therefore, there are 135 male employees in the company.
Question 19
Question
If xis directly proportional to yand inversely proportional to z, and x= 12
when y= 6 and z= 4, find xwhen y= 10 and z= 5.
11
Solution
Given that xis directly proportional to yand inversely proportional to z, we
can write:
x∝y
z
This implies that there exists a constant ksuch that:
x=ky
z
We are given that x= 12 when y= 6 and z= 4, so we can find the value of k:
12 = k6
4
k= 8
Step 1: Using the value of k, write an equation relating x,y, and z:
x= 8y
z
Step 2: Substitute y= 10 and z= 5 into the equation to find x:
x= 810
5
x= 8 ×2
x= 16
Therefore, when y= 10 and z= 5, x= 16.
Question 20
Question
A recipe calls for 3 cups of flour and 2 cups of sugar to make a certain dessert.
If you want to make 5 batches of the dessert, how many cups of sugar will you
need?
Solution
Let xrepresent the number of cups of sugar needed to make 5 batches of the
dessert.
Step 1: Set up a proportion based on the ratio of cups of flour to cups of
sugar in the recipe.
3 cups of flour
2 cups of sugar =15 cups of flour
x
12
Step 2: Cross multiply and solve for x.
3x= 2 ×15
3x= 30
x= 10
Step 3: Therefore, to make 5 batches of the dessert, you will need 10 cups
of sugar.
Question 21
Question
If 3 liters of a solution contain 15g of salt, how many grams of salt are there in
9 liters of the solution?
Solution
Step 1: Find the ratio of salt to solution in the given situation. Let xrepresent
the grams of salt in 9 liters of the solution. We know that the ratio of salt to
solution is the same in both cases, so we can set up the following proportion:
15 grams
3 liters =xgrams
9 liters
Step 2: Solve for xusing the proportion. Cross-multiplying gives:
15 ×9=3x
Step 3: Calculate the value of x.
135 = 3x
x=135
3
x= 45
Therefore, there are 45 grams of salt in 9 liters of the solution.
Question 22
Question
A group of students decided to share the cost of a 12-serving cake equally. If 5
students did not show up, each of the remaining students had to pay 1.5 times
the original amount. How many students did not show up?
13
Solution
Step 1: Let xbe the original cost per student when all students show up, and
let nbe the total number of students.
Step 2: Initially, the cost per student is x. Therefore, the total cost of the
cake is 12x.
Step 3: When 5 students did not show up, the number of students who
shared the cake became n−5. Each of the remaining students had to pay 1.5x.
Step 4: The new total cost of the cake is (n−5)(1.5x).
Step 5: Since the total cost of the cake remains the same whether all students
show up or not, we can set up an equation:
12x= (n−5)(1.5x)
Step 6: Simplifying the equation gives:
12x= 1.5nx −7.5x
Step 7: Rearranging the equation gives:
12x+ 7.5x= 1.5nx
Step 8: Combining like terms gives:
19.5x= 1.5nx
Step 9: Dividing both sides by xgives:
19.5=1.5n
Step 10: Finally, solving for ngives:
n=19.5
1.5= 13
Step 11: Therefore, the total number of students initially is 13, and the
number of students who did not show up is 5.
Question 23
Question
If 5 men can complete a construction project in 12 days, how many days will it
take for 8 men to complete the same project?
14
Solution
Step 1: Let’s denote the number of days it would take for 8 men to complete
the project as d.
Step 2: We can set up a proportion based on the number of men and the
number of days taken to complete the project:
5 men
12 days =8 men
ddays
Step 3: Cross-multiply to solve for d:
5·d= 12 ·8
Step 4: Simplify and solve for d:
5d= 96
d=96
5= 19.2
Step 5: Therefore, it will take 8 men approximately 19.2 days to complete
the construction project.
Question 24
Question
Solve for x:3
x−1=2
x+2
Solution
Step 1: Cross multiply to eliminate the denominators:
(3)(x+ 2) = (2)(x−1)
Step 2: Expand both sides:
3x+ 6 = 2x−2
Step 3: Move all terms involving xto one side:
3x−2x=−2−6
Step 4: Simplify the equation:
x=−8
15
Step 5: Check the solution by substituting x=−8 back into the original
equation: 3
(−8) −1=2
(−8) + 2
3
−9=2
−6
−1
3=−1
3
Since the left and right sides are equal, x=−8 is the correct solution.
Question 25
Question
Solve the following proportion for x:2
5=x−3
4x+1 .
Solution
We can solve the proportion by cross multiplying.
Step 1: Cross multiply to get 2(4x+ 1) = 5(x−3).
2(4x+ 1) = 5(x−3)
8x+ 2 = 5x−15
Step 2: Subtract 5xfrom both sides and add 15 to both sides.
8x+ 2 = 5x−15
8x−5x=−15 −2
3x=−17
Step 3: Divide by 3 to solve for x.
3x=−17
x=−17
3
Step 4: Simplify the result.
x=−17
3
16
Question 26
Question
Simplify the following ratio and express it in the form a:b, where aand bare
integers with no common factors other than 1:
2×53×7
32×5×112
Solution
To simplify the given ratio, we will first simplify the numerator and denominator
separately and then divide them to get the final ratio.
Step 1: Simplify the numerator
2×53×7=2×125 ×7 = 1750
Step 2: Simplify the denominator
32×5×112= 9 ×5×121 = 45 ×121 = 5445
Step 3: Divide the simplified numerator and denominator to find
the ratio
1750
5445 =350
1089 =50 ×7
33 ×33 =50 ×7
3×11 ×3×11 =50
3×11×7
3×11 =50
33×7
33 =50 ×7
33 ×33 =50
33
Question 27
Question
If two numbers are in the ratio 3 : 5 and their sum is 128, find the two numbers.
Solution
Step 1: Let the two numbers be 3xand 5x, where xis a common multiplier.
Step 2: Since their sum is 128, we can write the equation:
3x+ 5x= 128
Step 3: Simplify the equation to find the value of x:
8x= 128
Step 4: Divide both sides by 8 to solve for x:
x=128
8
17
x= 16
Step 5: Now, we can find the two numbers by substituting x= 16 back into
3xand 5x: The two numbers are:
3(16) = 48
5(16) = 80
Step 6: Therefore, the two numbers are 48 and 80.
Question 28
Question
If x:y= 3 : 4 and y:z= 5 : 2, what is the ratio of x:y:z?
Solution
To find the ratio of x:y:z, we need to combine the given ratios x:yand y:z.
Step 1: Determine a common term to connect the two ratios.
Since yis in both ratios, we can use it as the connecting term.
Step 2: Use the given ratios to rewrite x:yand y:z.
From the given ratios:
x:y= 3 : 4
y:z= 5 : 2
Step 3: Combine the two ratios to find x:y:z.
Given that x:y= 3 : 4 and y:z= 5 : 2, we can combine them to find
x:y:z:
x:y:z=3:4:2
Therefore, the ratio of x:y:zis 3 : 4 : 2 .
Question 29
Question
If 3 liters of a solution contains 15
18
Solution
Step 1: Let’s first determine how much salt is in the initial solution of 3 liters.
Step 2: Calculate the amount of salt in the initial solution. Step 3: Set up a
proportion to find the amount of water needed to dilute the solution. Step 4:
Solve the proportion to find the amount of water required.
Step 1: Determine the amount of salt in the initial solution. Let xbe the
amount of salt in 3 liters of the initial solution.
Step 2: Calculate the amount of salt in the initial solution. Since the initial
solution is 15
x= 0.15 ×3 = 0.45 liters
Now, we need to reduce the concentration of salt to 10
Step 3: Set up a proportion to find the amount of water needed to dilute
the solution. Let ybe the amount of water needed to dilute the solution.
We have the proportion: 0.45
3 + y= 0.10
Step 4: Solve the proportion to find the amount of water required. Cross
multiply to solve the proportion:
0.45 = 0.10(3 + y)
0.45 = 0.30 + 0.10y
0.10y= 0.15
y= 1.5 liters
Therefore, 1.5 liters of water must be added to the solution to reduce the
concentration of salt to 10
Question 30
Question
If a car travels 210 miles in 3 hours, how long will it take for the car to travel
560 miles?
Solution
Let xrepresent the time (in hours) it will take for the car to travel 560 miles.
We can set up a proportion to solve for x.
Step 1: Set up the proportion using the given information.
210
3=560
x
19
Step 2: Cross multiply to solve for x.
210x= 3 ×560
210x= 1680
Step 3: Solve for x.
x=1680
210
x= 8
Step 4: Final Answer: The car will take 8 hours to travel 560 miles.
Question 31
Question
Solve for xin the proportion 3
x+4 =x
6.
Solution
Step 1: Multiply both sides of the equation by 6(x+ 4) to eliminate the denom-
inators. 3
x+ 4 ·6(x+ 4) = x
6·6(x+ 4)
18 = x(x+ 4)
Step 2: Expand the right side of the equation.
18 = x2+ 4x
Step 3: Rearrange the equation into standard form.
x2+ 4x−18 = 0
Step 4: Solve the quadratic equation by factoring or using the quadratic
formula.
x2+ 6x−2x−18 = 0
x(x+ 6) −2(x+ 6) = 0
(x−2)(x+ 6) = 0
Step 5: Set each factor to zero and solve for x.
x−2 = 0 or x+ 6 = 0
x= 2 or x=−6
20
Step 6: Check the solutions to ensure they are valid in the original propor-
tion. When x= 2: 3
2+4 =2
6
3
6=2
6
1
2=1
3
When x=−6: 3
−6+4 =−6
6
3
−2=−1
This solution is extraneous.
Therefore, the solution to the proportion is x= 2.
Question 32
Question
If 4 men can build a wall in 10 days, and 6 women can build the same wall in
8 days, how many days will it take for 2 men and 3 women to build the wall
together?
Solution
Step 1: Calculate the rate at which each man and each woman can build the
wall. Let xbe the number of days for 1 man to build the wall, and ybe the
number of days for 1 woman to build the wall. Using the given information: -
4 men can build the wall in 10 days, so the rate for each man is 1
4·10 =1
40 walls
per day. - 6 women can build the wall in 8 days, so the rate for each woman is
1
6·8=1
48 walls per day.
Step 2: Set up the ratio and proportion equation. Let Dbe the number of
days for 2 men and 3 women to build the wall together. The combined rate for
2 men and 3 women working together is:
2·1
40 + 3 ·1
48 =1
20 +1
16 =9
160 walls per day
Using the formula for work, rate ×time = work, we have:
9
160 ×D= 1
9D
160 = 1
21
Step 3: Solve for the number of days, D. Multiplying both sides by 160, we
get:
9D= 160
D=160
9≈17.78
Therefore, it will take approximately 17.78 days for 2 men and 3 women to
build the wall together.
Question 33
Question
A recipe for a fruit salad calls for 2 cups of blueberries for every 3 cups of
strawberries. If we have 8 cups of blueberries, how many cups of strawberries
should we use to maintain the same ratio of blueberries to strawberries?
Solution
Step 1: Determine the ratio of blueberries to strawberries in the original recipe.
Let the number of cups of blueberries be 2xand the number of cups of straw-
berries be 3x, where xis a constant representing the ratio. Thus, the ratio of
blueberries to strawberries in the original recipe is 2x: 3xor 2 : 3.
Step 2: Calculate the number of cups of strawberries needed to maintain the
same ratio with 8 cups of blueberries. Since we have 8 cups of blueberries, set
up a proportion to find the number of cups of strawberries needed:
8 cups of blueberries
2=xcups of strawberries
3
4 = x
3
x= 12
Therefore, we need 3×12 = 36 cups of strawberries to maintain the same ratio
with 8 cups of blueberries.
Question 34
Question
Solve the following proportion for x:
3
x=x+ 4
8
22
Solution
To solve the proportion, we cross-multiply the terms to eliminate the fractions
and then solve for x.
Step 1: Cross-multiply the terms in the proportion:
3·8 = x·(x+ 4)
Step 2: Simplify both sides of the equation:
24 = x2+ 4x
Step 3: Rearrange the equation into a quadratic form:
x2+ 4x−24 = 0
Step 4: Factor the quadratic equation:
(x+ 6)(x−4) = 0
Step 5: Set each factor to zero and solve for x:
x+ 6 = 0 or x−4 = 0
Step 6: Solve for x: If x+ 6 = 0, then x=−6. If x−4 = 0, then x= 4.
Step 7: Check the solutions by substituting back into the original equation:
For x=−6: 3
−6=−6+4
8
1
−2=−2
8
−1
2=−1
4
This solution is not valid.
For x= 4: 3
4=4+4
8
3
4=8
8
3
4= 1
This solution is valid.
Therefore, the solution to the proportion is x= 4.
Question 35
Question
Solve the following proportion for x:3
2x+4 =x+1
5
23
Solution
To solve the proportion, we will cross multiply and simplify the resulting equa-
tion.
Step 1: Cross multiply to get rid of the fractions.
(3)(5) = (2x+ 4)(x+ 1)
Step 2: Expand both sides of the equation.
15 = 2x2+ 2x+ 4x+ 4
Step 3: Combine like terms on the right side.
15 = 2x2+ 6x+ 4
Step 4: Rearrange the equation to set it equal to zero.
2x2+ 6x+ 4 −15 = 0
2x2+ 6x−11 = 0
Step 5: Use the quadratic formula to solve for x.
x=−6±p62−4(2)(−11)
2(2)
x=−6±√36 + 88
4
x=−6±√124
4
x=−6±2√31
4
x=−3±√31
2
So, the solutions for xare x=−3+√31
2and x=−3−√31
2.
24
