Percent, Ratios, and Proportions

Question 1

Define:

ratio

Answer 1

A ratio is a comparison of two numbers.

Example: 3 to 5

Question 2

What are the three ways to express a ratio of two numbers?

Answer 2

A ratio can be expressed in three different ways:

• 4 to 5
• 4 : 5
• ⁴/₅

Question 3

One of the ways to express a ratio is to write it as a fraction.

So, are ratios a lot like fractions?

Answer 3

Yes, ratios are a lot like fractions.

²/₃ = ⁴/₆ = ⁶/₉ = ²⁰⁰/₃₀₀ - these are equivalent fractions but also equal ratios.

The fraction doesn’t change if both the numerator and the denominator are multiplied by the same number. So,

²/₃ = ⁴/₆ = … = ^2x/_3x where x is a whole number.

Question 4

How are ratios different from fractions?

Answer 4

A fraction shows part (the numerator) of the whole (the denominator).

A ratio can show a comparison between two parts of the same whole as well as a comparison between one part and the whole.

Example:

The ratio of cats to dogs at a pet shop is 2 to 3. At the very least, there are 5 (2 + 3) pets at this pet shop.

So, the ratio of cats to dogs is ²/₃. The ratio of cats to all pets in the store is ²/₅. The ratio of dogs to all pets in the store is ³/₅.

Question 5

Define:

part-to-part ratio

Answer 5

A part-to-part ratio represents a comparison between parts of a whole.

Example:

The ratio of girls to boys in a class is 3 to 5.

The ratio of plain bagels to poppy seed bagels to everything bagels is 5 : 4 : 3.

Question 6

Define:

part-to-whole ratio

Answer 6

A part-to-whole ratio represents a comparison between one part of the ratio and a whole.

Example:

The ratio of boys to all students in the class is 3 to 8. The ratio of girls to all students in the class is 5 to 8.

Question 7

How do you convert from a part-to-part ratio to a part-to-whole ratio?

Answer 7

A part-to-part ratio represents parts of a whole. To find the “whole”, add the “parts” together.

Example:

The ratio of girls to boys in a class is 3 to 5. This is a part-to-part ratio. Add the “parts” to find the “whole”. 3 + 5 = 8.

In other words, ³/₈ of all students are girls and ⁵/₈ of all students are boys.

Question 8

Can you find the ratio of two numbers if you know the number of each item?

Answer 8

Of course, you can find the ratio if you know the actual number of each item.

Example:

There are 400 fiction books and 600 non-fiction books in a library. What is the ratio of fiction books to non-fiction books?

Reduce ⁴⁰⁰/₆₀₀ fraction to its lowest term. The ratio of fiction books to non-fiction books is 2 : 3.

Question 9

What can you find if all you are given is the ratio of items?

Answer 9

You can find a number that the total number of items must be a multiple of.

Example:

The ratio of jeans to the tee shirts in your closet is 2 to 5.

²/₅ = ^2x/_5x. x could be any whole number. At the very least, there are 7 (2 + 5) jeans and tee shirts in your closet, but could also be 14 (4 + 10), 21 (6 + 15) or any multiple of 7.

Question 10

If all you are given is a ratio of two items, can you find:

• the number of each item
• the total number of items?

Answer 10

No, you cannot find the number of each item or the total number of items.

Example:

The ratio of juniors to seniors at a party is 5 to 2. How many people are at the party? How many juniors? How many seniors?

All you can determine from the given is that the total number of people has to be a multiple of 7. The number of juniors has to be a multiple of 5 while the number of seniors has be a multiple of 2.

Question 11

The ratio of juniors to seniors at a party is 5 to 2. There are 24 seniors at the party.

• Can you find the number of juniors?
• Can you find the total number of people at the party?

Answer 11

Yes, you can find both the number of juniors and the total number of people.

^J/_S = ⁵/₂ = ^5x/_2x = ^?/₂₄

S = 24 = 2x ⇒ x = 12

J = 5 x 12 = 60

Total = J + S = 60 + 24 = 84

Question 12

If you are given a ratio of items and the actual number of one part of the ratio, what can you determine?

Answer 12

You can determine the actual number of the other part of the ratio as well as the total number of items.

Example:

The ratio of boys to girls in your class is 5 to 3 and there are 18 girls.

⁵/₃ = ^5x/_3x = ^?/₁₈

3x = 18 ⇒ x = 6 ⇒ 5x = 30

30 + 18 = 48 - total number of students.

Question 13

If you have two part-to-part ratios with a common part,

the ratio of a to b is 2 to 5

and

the ratio of b to c is 5 to 7,

what is the ratio of a to c?

Answer 13

The ratio of a to c is 2 to 7.

Question 14

Define:

rate

Answer 14

A rate is a ratio between two measurements.

Normally, the two terms of a rate are measured in different units.

Example: Miles per hour is the rate of speed.

Question 15

What is the formula for the rate of speed of an object?

Answer 15

The rate of speed can be found by dividing distance by time.

R = ^D/_T ⇒ D = R x T

Example:

A car drives 100 miles in 2 hours. What is its rate of speed?
¹⁰⁰/₂ = 50 mph

Question 16

You walk from home to school with an average speed of 6 mph. On the way back from school your average speed is 4 mph.

What is your average speed for the entire trip?

Answer 16

The average speed for the entire trip is not (4+ 6) ÷ 2 = 5 mph! It equals 4.8 mph.

Avg Speed = ^{Total Distance}/_{Total Time}

D is the distance in each direction; therefore, the entire distance is 2D.

T1 is your time on the way from home to school. T2 is your time on the way back. The total time is the sum of T1 and T2.

T1 = ^D/_S1, T2 = ^D/_S2

Avg Speed = 2D/(T1 + T2) = 2D/(D/S1 + D/S2) = 2S1 * S2/ (S1 + S2)

Avg Speed = 2 * 6 * 4/ (6 + 4) = 4.8 mph

Question 17

What is the formula for the average rate of speed?

Answer 17

Average speed is defined as total distance divided by total time.

Average Speed = ^{Total Distance}/_{Total Time}

Question 18

Define:

proportion

Answer 18

Two equal ratios are called a proportion.

a : b = c : d

Example: ⁴/₇ = ¹²/₂₁

Question 19

How do you check if two ratios are equal?

¹/₃ = ³³/₉₉

Answer 19

A common way to check if two ratios are equal is to cross-multiply the numbers making up those ratios.

¹/₃ = ³³/₉₉

The product of 1 and 99 equals the product of 3 and 33; therefore, this proportion is a true proportion.

Question 20

What method do you use to solve a proportion?

Answer 20

When solving a proportion, use the cross-multiplication method.

^a/_b = ^c/_d

a * d = b * c

Multiply the numerator of the fraction on the left side of the equation by the denominator of the fraction on the right side. Repeat for the other denominator and numerator.

Question 21

When are two quantities in direct proportion?

Answer 21

Two quantities, A and B, are in direct proportion if both quantities change by the same factor.

Example:
1 can of soda costs $0.50. It would cost you $1.00 to buy 2 cans. For 6 cans you pay $3.00; for 12 cans you would pay $6.00. Notice that changing the number of cans you buy will change the total amount of money you pay. It changes by the same factor, 2.

Question 22

If it takes 5 men 12 days to build a house, how many days would it take 6 men to build the same house if they work at the same rate?

Use simple logic to solve this problem.

Answer 22

Logically, the more men are on the job, the less time it’s going to take for the job to be completed. This problem is an example of an indirect (or inverse) proportion.

A complete job would take 5 men * 12 days = 60 “men-days”. Divide that total by the new number of workers. It would take 6 men 10 days to build the same house.

Or you can set up a proportion with two ratios and take the reciprocal of one of them:

5 men → 12 days
6 men → x days

⁵/₆ = ^x/₁₂ ⇒ 6x = 60 ⇒ x = 10

Question 23

When are two quantities in an indirect (or inverse) proportion?

Answer 23

Two quantities, A and B, are in indirect (inverse) proportion if by whatever factor A changes, B changes by a reciprocal of that factor.

Example:
When quantity A doubles, quantity B becomes half as large.
The terms exchange; the inverse of the ratio A/B is the ratio B/A.

Question 24

What is the formula for indirect (inverse) proportions?

Answer 24

The formula for indirect proportions is:

x₁ : x₂ = y₂ : y₁

Cross-multiply:

x₁ * y₁ = x₂ * y₂

Question 25

Define:

percentage

Answer 25

A percentage shows the ratio of a number to 100. It is denoted by the following symbol: %.

A percentage is a part of a whole (like a fraction or a decimal), expressed in hundredths.

Example: 2% = 0.02

Question 26

What is 1%?

Answer 26

1% is the fraction ¹/₁₀₀.

“Per cent” literally means “per hundred”.

Question 27

What is x% of a number?

Answer 27

x% of a number is ^x/₁₀₀ of that number.

Question 28

How can you prove that x% of y always equals y% of x?

Answer 28

x% of y = ^x/₁₀₀ * y = ^xy/₁₀₀

y% of x = ^y/₁₀₀ * x = ^yx/₁₀₀

Two fractions are equal and, therefore,

x% of y equals y% of x.

Question 29

How do you convert fractions into percentages?

Answer only for fractions whose denominators are factors of 100.

Answer 29

If the denominator of a fraction is a factor of 100, change the fraction to an equivalent fraction with a denominator of 100. The numerator of this equivalent fraction shows you the percentage.

Example: ³/₅ = ⁶⁰/₁₀₀ = 60%

Question 30

How do you convert fractions into percentages?

Answer only for fractions whose denominators are not factors of 100.

Answer 30

If the denominator of a fraction is not a factor of 100, divide the numerator by the denominator, then multiply by 100.

Example:
⁴/₉ = 4 ÷ 9 = 0.4444 x 100 = 44.44%
¹/₈ = 0.125 x 100 = 12.5%

Question 31

How do you convert percentages into fractions?

Answer 31

Convert the percentage into a fraction with a denominator of 100.

75% = ⁷⁵/₁₀₀ = ³/₄

12.5% = ^12.5/₁₀₀ = ¹²⁵/₁₀₀₀ = ¹/₈

Question 32

How do you convert percentages into decimals?

Answer 32

Write the percentage as a fraction with a denominator of 100, then express the fraction as a decimal.

35% = ³⁵/₁₀₀ = 0.35

Or, move the decimal point two places to the left.

225% = 2.25

Question 33

How do you convert decimals into percentages?

Answer 33

First, change the decimal into a fraction, then convert the fraction into a percentage.

0.5 = ⁵⁰/₁₀₀ = 50%

Or, simply move the decimal point two places to the right.

0.3478 = 34.78%

Question 34

Which three methods can be used to find a percent of a number?

Answer 34

1. Fraction Method
2. Decimal Method
3. Proportion Method

Question 35

How do you use the fraction method to find a percent of a number?

Answer 35

Change the percent to an equivalent fraction and simplify if possible. Then multiply by the number in question.

Example:
Find 40% of 60.
40% = ⁴⁰/₁₀₀ = ⁴/₁₀ = ²/₅
60 x ²/₅ = ¹²⁰/₅ = 24

Question 36

How do you use the decimal method to find a percent of a number?

Answer 36

Change the percent to an equivalent decimal. Then multiply by the number in question.

Example:
Find 40% of 60.
40% = 0.4
60 x 0.4 = 24

Question 37

What formula can you use to express a percentage in terms of two given numbers?

Answer 37

Percent = ^Part/_Base * 100% ⇒

You can write this formula as a proportion:

^Part/_Base = ^Percent/₁₀₀

“Part” and “base” are two given numbers. “Base” is the total amount of something. “Part” is the partial amount of something.

Question 38

Write the percentage formula as a proportion.

Answer 38

^Part/_Base = ^Percent/₁₀₀

A percent is a ratio of a number to 100. The idea of a proportion is that this ratio could equal any other ratio, according to the problem.

Example:

25 is 40% of some number. Write this statement as a proportion.

According to the definition, 40% is a ratio of ⁴⁰/₁₀₀. To write a proportion, we need another equal ratio. 25 is a part of something bigger since it represents only 40% of the total. So, ²⁵/_Base is our second ratio.

²⁵/_Base = ⁴⁰/₁₀₀

Question 39

How do you use the proportion method to find what percent is one number of another number?

Answer 39

Percent = ^{Part * 100}/_Base

Write the following proportion, cross-multiply and solve for percent.

^Is/_Of = ^Part/_Base = ^Percent/₁₀₀

Example:

20 is what percent of 80?

²⁰/₈₀ = ^Percent/₁₀₀

Percent = ^{20 x 100}/₈₀ = 25%

Question 40

Using the proportion method, you can find any of the three variable components of the proportion formula: part, base, and percent.

If you know the percent and the “part”, how do you find the “base”?

Answer 40

Base = ^{Part x 100}/_Percent

^{Cross-multiply and solve for the “base”.}

^Part/_Base = ^Percent/₁₀₀

Example:

16 is 20% of what number?

¹⁶/_Base = ²⁰/₁₀₀

“Base” = ^{16 x 100}/₂₀ = 80

Question 41

Using the proportion method, you can find any of the three variable components of the proportion formula: part, base, and percent.

If you know the percent and the “base”, how do you solve for the “part”?

Answer 41

Part = ^{Base x Percent}/₁₀₀

_{Cross-multiply and solve for the “part”.}

^Part/_Base = ^Percent/₁₀₀

Example:

20% of 80 is ….?

^Part/₈₀ = ²⁰/₁₀₀

“Part” = ^{80 x 20}/₁₀₀ = ¹⁶⁰⁰/₁₀₀ = 16

Question 42

Let’s say you had $100 last year. Now you have $250 because you got a generous gift for your birthday. How does the current amount of money you have compare to the amount you had last year?

Answer 42

You can say that you have $150 more than last year. But to really understand how the value changed, you need to find the ratio of the difference of the current $ amount and the $ amount last year to the $ amount you had last year.

250 - 100 = 150

¹⁵⁰/₁₀₀ = 1.5

In percent, it’s 1.5 x 100% = 150%

150% is the percent change.

Question 43

How do you understand percent change?

Answer 43

When the value of something changes over time, we use percent change to understand how the new value compares to the old value.

Percent increase (gain) and percent decrease (loss) are measures of percent change.

Question 44

What is the formula for percent change?

Answer 44

Formula for percent change:

Question 45

How do you find an increase in value by a certain percentage?

Answer 45

To increase a number by a percent, add the percent to 100%, convert to a decimal, and multiply by that number.

Increased Value =

(100% + % increase) x Original Value

Example:
Increase 50 by 50%.
100% + 50% = 150%. Convert 150% to 1.5, then multiply by 50: 1.5 x 50 = 75

Question 46

How do you find a decrease in value by a certain percentage?

Answer 46

To decrease a number by a percent, subtract the percent from 100%, convert to a decimal, and multiply.

Decreased Value =

(100% - % decrease) x Original Value

Example:
Decrease 50 by 20%
100% - 20% = 80%. Convert 80% to 0.8 and multiply by the original amount: 50 x 0.8 = 40

Question 47

What is the original price of a pair of shoes if the sale price is $70 after a 30% discount?

Answer 47

Re-phrasing the problem, the original price (always 100%) was reduced by 30% to become the sale price of $70. Logically, this means that 70 is 70% of the original price.

70% of x = 70 ⇒ 0.7x = 70 ⇒ x = 100

Or you can set up a proportion relating the sale price and the percent discount to the original price (100%).

100% - 30% = 70%

If 70% is $70, then 100% is x dollars?

⁷⁰/_x = ⁷⁰/₁₀₀ ⇒ x = 100

Question 48

How do you find the original value, before it was increased or decreased?

Answer 48

The original value is 100%. You can set up a proportion relating the new value and the percent change to 100%.

Example:

After a 25% increase, the population of town A became 60,000. What was the original population of town A?

125% is 60,000
100% is ?

Set up a proportion: ¹²⁵/₁₀₀ = ^60,000/_?

Or you could go straight to writing an equation. 1.25x = 60,000 ⇒ x = 48,000

Question 49

What is interest in terms of borrowing money?

What is interest rate?

Answer 49

Interest is the cost of borrowing money.

Interest rate is the cost of borrowing money expressed in percent of the borrowed amount.

Question 50

How much money would you make in interest after 2 years if you put $100 in the bank at a 2% simple interest rate?

Answer 50

If you make 2% a year, in one year you would make $2 because 2% of 100 is 2.

However, you decide to keep your money at this bank for another year.

After 2 years you will make $4 in interest.

Question 51

What is simple interest?

Answer 51

Simple interest is interest that is calculated on original principal only.

Question 52

What is the formula for simple interest?

Answer 52

Simple interest = P * r * t

P is principle (original amount of the investment)
r is interest rate for one period
t is time (the number of time periods)

Question 53

If you borrow $600 for 2 years at a 10% interest rate compounded semi-annually, what is total interest you earned over two years?

Answer 53

The interest earned in each period is added to the principal of the previous period to become the principal for the next period.

Semi-annually means twice a year. So, all together, there are four compounding periods.

• interest 1st period: p * r * t = 600 * 0.1 * 1 = 60
• Interest 2nd period: (600 + 60) * 0.1 * 1 = 66
• Interest 3rd period: (660 + 66) * 0.1 * 1 = 72.6
• Interest 4th period: (726 + 72.6) * 0.1 * 1 = 79.9

Total interest earned over two years =

60 + 66 + 72.6 + 79.9 = 278.5

Question 54

What does the term “compound interest” mean?

Answer 54

“Compound interest” is interest that is calculated each period on both the original principal and on all interest accumulated in the past period.

Question 55

What is the formula for compound interest?

Answer 55

Compound interest = P(1 + ^r/_n)^nt

P is original principle
r is interest rate
n is the number of times per year the interest is compounded
t is the number of years the money is invested

Question 56

What is 0.005 expressed as a percentage?

Answer 56

0.50%

Multiply by 100, or simply move the decimal point two places to the right.

Question 57

What is 165% written as a decimal number?

Answer 57

1.65

Divide by 100, or simply move the decimal point two places to the left.

Question 58

The attendance of a class in your school is 87.5%. What is the attendance of this class expressed as a fraction?

Answer 58

⁷/₈

Move the decimal point two places to the left to convert 87.5% into a decimal…. 0.875. Write it as a fraction ⁸⁷⁵/₁₀₀₀ and reduce to its lowest terms.

Question 59

What is ²/₅ expressed in percent?

Answer 59

40%

Convert ²/₅ to an equivalent fraction with a denominator of 100.

²/₅ = ⁴/₁₀ = ⁴⁰/₁₀₀

Then convert ⁴⁰/₁₀₀ to a percentage.

Question 60

What is the relationship between a percentage, a fraction, and a decimal?

Answer 60

A percentage is a fraction or a decimal, expressed in hundredths.

Question 61

If you have two numbers and you want to know what percent one number is of another given number, what do you do?

Answer 61

1,000%

Determine which number is the “part” and which number is the “base” and write a proportion.

^Is/_Of = ^Part/_Base = ^Percent/₁₀₀

Percent = ^{Part x 100}/_Base

Example:

What percent of 30 is 300?

³⁰⁰/₃₀ = ^Percent/₁₀₀

Percent = ^{300 x 100}/₃₀ = 1,000%

Question 62

15 of the 25 people in your class are applying to college. What percentage of your class is applying to college?

Answer 62

60%

Which number is the “part”? Which number is the “base”? 15 is a part of the total class, so 15 is the “part”, and 25 is the “base”.

Write a proportion.

^Part/_Base = ^Percent/₁₀₀

¹⁵/₂₅ = ^x/₁₀₀

Cross-multiply and solve for x. x = 60%

Question 63

Calculate percentages below and find the difference between them:

I. What percent is 20 of 400?

II. What percent is 400 of 20?

percent 2 - percent 1 = ?

Answer 63

Percent 2 - percent 1 = 1,995%

Use the proportion method:

^Part/_Base = ^Percent/₁₀₀

Percent = ^{Part x 100}/_Base

I. What percent is 20 of 400?

²⁰/₄₀₀ = ^Percent/₁₀₀

Percent = ^{20 x 100}/₄₀₀ = 5%

II. What percent is 400 of 20? Reverse the numbers in the formula.

⁴⁰⁰/₂₀ = ^Percent/₁₀₀

Percent = ^{400 x 100}/₂₀ = 2,000%

The difference is 2000% - 5% = 1,995%

Question 64

What is 400% of 5?

Answer 64

20

400% of 5 is a number that is 4 times greater than 5.

You can set up a proportion to prove it.

^Part/_Whole = ^Percent/₁₀₀

^Part/₅ = ⁴⁰⁰/₁₀₀ ⇒ ^Part/₅ = 4 ⇒

Part = 5 x 4 = 20

Question 65

6000% + 600% + 60% + 6% = ?

(a) 6,666
(b) 666.6
(c) 66.66
(d) 6.666

Answer 65

(c) 66.66

Have you noticed that the answers are not in percentages? Make sure you convert from percents into numbers.

The sum of the percentages is 6,666%. To convert, mentally divide the sum by 100, or just move the decimal two places to the left.

Question 66

What is m% of n expressed as a fraction?

Answer 66

m% of n expressed as a fraction is

^m/₁₀₀ * n = ^mn/₁₀₀

Question 67

48% of 72 = ? of 48

Answer 67

72%

Remember

x% of y always equals y% of x.

Question 68

In your class there are 12 boys and 26 girls.

• What is the ratio of girls to boys?
• What is the ratio of boys to the total number of students?

Answer 68

^G/_B = ²⁶/₁₂ = ¹³/₆

^B/_Total = ¹²/₃₈ = ⁶/₁₉

Always simplify fractions if possible.

Question 69

Your friend biked 18 miles in 3 hours. You biked 24 miles at the same speed. How long was your bike ride?

Answer 69

Your bike ride was 4 hours.

First, figure out your friend’s speed using the rate of speed formula

Rate of Speed = ^Distance/_Time

R = ¹⁸/₃ = 6 mph

Using the same formula, solve for T given the distance 24 miles and the speed 6 mph:

T = ^D/_R = ²⁴/₆ = 4 hours

Question 70

Last summer you grew from 60” to 66”. By what percentage did your height increase?

Answer 70

Your height increased by 10%.

Use the percent change formula.

The change in your height is 6”. Your original height was 60”. Therefore, the percent change (or, in this case, increase) will be:

⁶/₆₀ x 100% = 10%

Question 71

The original price of an iPod is $120. If its sale price is $80, at what percentage has the iPod been discounted?

Answer 71

The iPod has been discounted by 33.3%.

Use the percent change formula.

The change in price is $40. The original price was $120. Therefore, the percent change (or, in this case, decrease) is:

⁴⁰/₁₂₀ x 100% = 33.3% (rounded up)

Question 72

The stock you invested in went up 200% in value. What is the new price of the stock if the original price was $20?

Answer 72

The new price of the stock is $60.

Use the formula for finding increased value:

Increased Value =

(100% + % Increase) x Original Value

(100% + 200%) x 20 = 60

or convert into fractions or decimals:

(1 + Increase) x Original Amount

(1 + 2) x 20 = 60

Question 73

A pair of jeans that normally costs $64 is on a 25% off sale. What is the discounted price of this pair of jeans?

Answer 73

The discounted price is $48.

Use the formula for finding the decrease in value:

Decreased Value =

(100% - % Decrease) x Original Value

(100% - 25%) x 64 = 48

or convert into fractions or decimals:

DV = (1 - Increase) x Original Value

(1 - 1/4) x 64 = 48

Question 74

A pair of boots was discounted from $149.99 to $99.99.

A sweater was discounted from $119.99 to $69.99.

Which item had a greater percent discount?

Answer 74

The sweater had a greater % discount.

Both items were discounted by $50. However, the original price of the boots was higher than the original price of the sweater.

⁵⁰/_149.99 * 100% < ⁵⁰/_69.99 * 100%

Remember, you need to relate the change in value to the original value.

Question 75

You put $400 in the bank at 5% interest for 5 years. Your friend puts in $1,000 at 6% interest for 2 years.

Who makes more money at the end of the term- you after 5 years or your friend after 2 years?

Answer 75

Unfortunately, your friend is going to make more money.

Use the formula for simple interest:

Interest = Rate x Principal x Time

Your interest = 0.05 x 400 x 5 = $100

Your friend’s interest = 0.06 x 1,000 x 2 = $120

Question 76

Two chocolate bars cost $5. How much would you have to pay for five of these bars?

Answer 76

5 bars would cost $12.50.

Write a proportion:

⁵/₂ = ^x/₅

Solve for x by cross-multiplying:

25 = 2x ⇒ x = 12.5

Question 77

The ratio of soccer players to basketball players in your class is equal to the ratio of tennis players to baseball players.

There are 12 soccer players, 10 basketball players, and 15 baseball players.

How many students play tennis?

Answer 77

18 students play tennis.

Write a proportion with the given ratios and cross-multiply to solve for the number of tennis players.

¹²/₁₀ = ^T/₁₅

10T = 180 ⇒ T = 18

Question 78

Let P, R, S and T be positive integers. The ratio of P to R is equal to the ratio of S to T. What is the smallest possible value of P if the product of R and S equals 18 and T is an odd number?

R x S = 18

T is odd

Answer 78

The smallest possible value of P is 2.

P : R = S : T

Use cross-multiplication to solve this proportion…..PT = RS = 18

Factor 18 into pairs: 1 x 18, 2 x 9, 3 x 6

Since T is odd, the possible choices are 3 and 9. The smallest value of P is 2 (which corresponds to T = 9).

Question 79

If 4 cans of soda cost $3.00, how much money do you pay for 6 cans?

Answer 79

6 cans of soda cost $4.50.

Use your knowledge of direct proportions, to set up one in this problem.

⁴/₆ = ³/_x

Cross-multiply and solve for x.

x = $4.50

Another way to solve the problem is to figure out the cost of 1 can of soda and multiply by 6.

Question 80

It takes 4 mice 6 days to eat 1 lb of cheese. How many days would it take for 10 mice to eat the same amount of cheese if they eat it at the same rate?

Hint: More is less…

Answer 80

This is an indirect relationship. It means that it would take less time for more mice to eat the cheese.

• Write down the ratio using one type of term (# of mice): 4 to 10
• Write down the ratio with the second term - days: 6 to x
• Invert (flip) one of the ratios.
• Solve as a direct proportion by cross-multiplying.

¹⁰/₄ = ⁶/_x ⇒ x = ^{4 x 6}/₁₀ = 2.4 days

Logically, 10 is 2.5 times more than 4. So, the time has to be 2.5 times less.

Question 81

There are 20 cars in the parking lot. The ratio of convertibles to the total number of cars is 2 to 5. How many convertible cars are there?

Answer 81

There are 8 convertible cars.

2 out of every 5 cars are convertibles. To find the number of convertible cars you can

• either multiply ²/₅ x 20 = 8
• or, write a proportion:

^Conv/₂₀ = ²/₅

Question 82

If the ratio of 2-door cars to 4-door cars on a parking lot is 3 to 8, which number could represent the total number of cars on the parking lot?

(a) 38
(b) 36
(c) 33
(d) 31

Answer 82

(c) 33

3 to 8 is a part-to-part ratio. Two parts add up to a “whole”.

3 + 8 = 11

The total number of cars on the parking lot has to be a multiple of 11. That number is 33.

Question 83

If there are 420 students in my school, then the ratio of boys to girls in my school cannot be

(a) 6 : 14
(b) 5 : 9
(c) 11 : 14
(d) 17 : 18

Answer 83

(c) 11 : 14

Answer choices are four part-to-part ratios. Add two “parts” together to find a “whole”. The total number of students has to be divisible by that number.

420 is divisible by 20, 14, and 35. It is not divisible by 25. Therefore, the ratio of boys to girls is not 11 to 14.

Question 84

If the ratio of 2-door cars to 4-door cars on a parking lot is 3 to 8, which number could not be the number of 4-door cars on the parking lot?

(a) 24
(b) 32
(c) 68
(d) 72

Answer 84

(c) 68

The number of 4-door cars on the parking lot has to be a multiple of 8. 68 is the only number in the answer choices that is not a multiple of 8.

Question 85

m is directly proportional to n. If m = 5 and n = ¹/₅, what is the value of m when n is 5?

Answer 85

m = 125

For n to change from ¹/₅ to 5, it had to be multiplied by 25. m and n are directly proportional; therefore, 5 (m) has to increase 25 times as well.

Question 86

In a pet store, the ratio of dogs to cats is 5 to 3. The ratio of dogs to fish is 2 to 25. What is the ratio of fish to cats?

D : C = 5 : 3

D : F = 2 : 25

Answer 86

F : C = 125 : 6

Let’s first answer what it is not. The ratio of fish to cats is not 25 : 3.

The ratio of dogs is the common part of two ratios; 5 in the first ratio and 2 in the second ratio. To compare two ratios, find the LCM of 5 and 2, then re-state the ratios with the common part expressed by the same number.

The ratio of fish to cats is 125 to 6.

Question 87

The ratio of milk chocolate bars to dark chocolate bars is 2 to 3. The ratio of dark chocolate bars to white chocolate bars is 2 to 5. What is the ratio of milk to white chocolate bars?

Answer 87

Milk : White = 4 : 15

If the ratios share a common part that is represented by different numbers, they are not proportional. You need to find the LCM of the numbers representing the common part and re-state the ratios.

M : D = 2 : 3 D : W = 2 : 5

The LCM of 3 and 2 is 6. Re-state the ratios.

M : D = 4 : 6 D : W = 6 : 15 ⇒

M : W = 4 : 15