TABE Level D Math: Number Theory

Number theory problems are about ratio, proportion, and percent.

A ratio is one way of comparing two numbers.

For example, the ratio of the width of a rug to its length may be 3 to 5. The width of the rug is of its length.

A proportion says two ratios are equal.

For example, is a proportion. This would say that a 3-by-5 rug is in the same proportion as a 9-by-15 rug. The two rugs are proportional in size. Their shapes are the same.

A percent is a ratio where one of the numbers is 100.

A 60-by-100 rug would have the same shape as a 3-by-5 rug or a 9-by-15 rug because and are all equal. Percents are used as ratios because they give a standard of comparison.

For example, if you wanted to know which is more square, a 3-by-5 rug or a 4-by-8 rug? You would calculate the width as a percent of the length in each case. If the width is 100% of the length, the rug is square. For the 3-by-5 rug you would say that the width is 60% of the length. For the 4-by-8 rug the width would only be 50% of the length. Therefore, the 3-by-5 rug is more square.

Most proportion problems ask you to find a number that makes two ratios equal. There are 4 numbers in a proportion, two numerators and two denominators. These problems tell you 3 of the numbers, and you have to find the fourth number.

In the percent problems you have to be able to convert between percents, fractions, and decimals. You also have to calculate percent increases and decreases. 

Change a Fraction to a Percent
To change a fraction to a percent, divide the numerator by the denominator and move the decimal point two places right (multiply by 100). This is just like finding the percent if you know the part and the whole.

Example A.

What percent of the square is shaded?

A. 3% B. 30% C. 60% D. 75%
The square is divided into 4 equal sections. Three parts are shaded, so the fraction that is shaded is . Divide 3 by 4 and move the decimal point two places to the right. The correct answer is d, 75%.

Example B.

A health club manager determines that 2 of 5 people who try out the club eventually join. What percent is this? A. 25% B. 30% C. 35% D. 40%
Because 2 of 5 eventually join, the fraction that joins is Divide 2 by 5 and move the decimal two places to the right. The correct answer is d, 40%.
 

Example C.

A large package of raisins contains 16 oz. A small package of raisings contains 4 oz. What percent of the large package is in the small package? A. 10% B. 25% C. 30% D. 40%
The small package contains of the amount of raisins in the large package. Divide 4 by 16 and move the decimal point two places to the right. The correct answer is b, 25%.

Change a Decimal to a Fraction
To change a decimal to a fraction, identify the place value of the digit farthest to the right. This tells you what the denominator of the fraction will be. Then write the digits in the decimal in the numerator.

Example D.

A can of soda contains .85 liters. Which of these is another way to show the amount of soda in a can of this size? A. B. C. D.
The digit farthest to the right is 5, and it is in the hundredths place. This makes the denominator 100. Therefore, the decimal .85 equals the fraction You can reduce this to by dividing the numerator and denominator by 5. The correct answer is c, .

Example E.

The average number of children in a certain U.S. city is 2.3. Which of these is another way to show the average number of children in this city? A. B. C. D.
The whole number part is 2. The digit farthest to the right is 3, and it is in the tenths place. This makes the denominator of the fractional part 10. Therefore, the decimal 2.3 equals the mixed number The correct answer is d.

Percent Problems
These are word problems that ask you to find the part, whole, or %. You learned how to do this in Lesson 5.
- To find the %, divide the part by the whole and move the decimal point 2 places to the right.
- To find the part, multiply the % by the whole, and move the decimal point 2 places to the left.
- To find the whole, divide the part by the percent, and move the decimal point 2 places to the right.

Example F.

A small New England town has 5000 registered voters. During a recent election, 2700 voted. What percent of registered voters actually voted in that election? A. 54% B. 52% C. 45% D. 27%
The problem asks for %. The part is 2700, and the whole is 5000. Use the formula and a calculator to get

Example G.

Suzanne spent $19.55 for dinner at a restaurant. She wants to leave a 15% tip. Which of the following is a way to calculating 15% of $19.55? A. $19.55 × .15 B. $19.55 × 1.5 C. $19.55 ÷ .15 D. $19.55 ÷ 1.5
In this problem, the tip is the part, the cost of dinner is the whole, and 15 is %. According to the formula the part is 19.55 × 15 with the decimal point moved two places to the left. The decimal point is after the 5 in the number 15. If you move that decimal point 2 places to the left, you get .15. The correct answer is a.
 

Example H.

George took a test with 20 questions on it. Four of his answers were wrong. What percent did he get right? A. 96% B. 86% C. 80% D. 75%
If 4 of his answers were wrong, 16 were right. The part is 16, and the whole is 20. Use the formula and a calculator to get . The correct answer is c.

Example I.

The retail price of a shirt is $30. The store will add 7% sales tax. How much will the shirt cost with tax? A. $30.70 B. $37.00 C. $32.10 D. $31.70
First find the amount of tax. This is the part. The whole is 30, and the % is 7. To calculator the part, multiply 7 times 30, and move the decimal point two places to the left. The amount of tax is $2.10. Add this to the $30.00 to get the total price $32.10, answer c.

Example J.

During the last six months, the price of gasoline went from $1.75 to $2.10. What was the percent increase in the price of gasoline? A. 17% B. 20% C. 35% D. 83%
The amount of the increase is $.35 ($2.10 − $1.75). The part is .35, and the whole is 1.75 (the starting amount). Use the formula and a calculator to get . The correct answer is b, 20%.

Example K.

Michelle paid $15 for a coffee pot that was on sale at 20% off. What was the original price? A. $18.00 B. $12.00 C. $18.75 D. $3.00
This is a tricky problem. You first have to realize that if the Michelle got 20% off the cost of the pot, she paid 80%. So the % is 80, not 20, the part is 15, and the problem asks you to find the whole. Use the formula and a calculator to get .

Solve a Proportion
Percent problems are proportions where the denominator of one fraction is 100:

In a proportion that is not a percent, other numbers replace the % and 100. These other two numbers are another part and whole.

In a proportion problem 3 of the 4 numbers are known, and you have to find the fourth number.

An example will help here. In the proportion problem you have to determine what number x is. In this example, you know the 2 denominators. Divide the larger one (8) by the example smaller one (4). Then multiply the numerator (3) by the answer (2), and that’s the other numerator . In the example, , and . If you knew the two numerators and only one denominator instead, the procedure would be exactly the same. 
 

Example L.

Michael made 6 free throws in 8 tries. How many would you expect him to make in 24 tries? A. 12 B. 16 C. 18 D. 20
Let x stand for the answer (number of free throws made in 24 tries). Then are the two equal fractions. First divide 24 by 8 and get 3. Then multiply 3 by 6 to get 18.

Example M.

The ratio of red tulips to yellow tulips in Cherie’s garden is 5:2. This year Cherie has 20 yellow tulips in her garden. How many red tulips does she have? A. 23 B. 25 C. 50 D. 60
The proportion in this problem is Twenty divided by 2 is 10, and 10 times 5 is 50.
The hair color of 500 members of the high school senior class was classified as brunette, blonde, or redhead. The results are shown in the table below.

Example N.

If there are 2000 students in the high school, how many brunettes would you expect to find?
Set up the proportion where x is the number of brunettes in the school. Then and . You would expect to find 920 brunettes in a school with 2000 students.

Number Theory Practice

1. A map has a scale of 1 inch = 5 miles. How far is it Barnesville from Carlton if the map distance between them is 3 inches? A. 15 miles B. 7 miles C. 9 miles D. 8 miles

2. A 12-ounce can of soda costs $1.25. How much would you expect a 15-ounce bottle to cost? A. $1.40 B. $1.56 C. $1.62 D. $1.75

3. Josh got 35% of the vote in a Student Council election at a school with 1500 students. If everyone at the school voted, how many votes did Josh receive? A. 500 B. 525 C. 540 D. 550

4. The cost of chicken wings at the local supermarket increased from $1.29 a pound to $1.59 a pound in the last month. About what percent increase is this? A. 19% B. 68% C. 30% D. 23%

5. Debra is planning to paint a picture of a photograph that is 4 inches wide by 6 inches high. If her painting must be 16 inches wide, how high should it be? A. 18 inches B. 20 inches C. 24 inches D. 28 inches

6. A large order of French fries costs $2.00. A super size order has 30% more fries. How much should a super size order of fries cost? A. $2.30 B. $2.60 C. $3.00 D. $3.30

7. George took a 20-question math test, and he got 16 problems right. How many problems could he expect to get right on a 50-question test covering the same topics? A. 46 B. 40 C. 38 D. 32

8. A 16-ounce package of frozen vegetables costs $2.10. How much would you expect an 8-ounce package to cost? A. $1.75 B. $1.45 C. $1.25 D. $1.05

9. A small package of 30 colored candies contained 10 red pieces. How many red pieces would you expect to find in a large package containing 45 pieces? A. 15 B. 20 C. 25 D. 30

10. Five of eight cameras sold in the Camera Mart Store are digital cameras. Last week Camera Mart sold 72 cameras. How many digital cameras did they sell? A. 36 B. 40 C. 45 D. 50

Answers: Number Theory Practice

1. a

2. b

3. b

4. d

5. c

6. b

7. b

8. d

9. a

10. c