# Fractions / Percentages

In doing problems that involve percentages, whether or not they also involve fractions, it’s important to remember that percentages are just fractions. For example, 87% is the same as 87/100, or, written as a decimal, it’s .87. When we do a problem that includes both fractions and percentages, it’s often easiest to convert the percentages to fractions or the fractions to percentages.

On this page we provide 3 examples of Fraction problems and 3 examples of Percentage problems.

Fraction Problems Example 1)

Harriet bought 120 fifty dollar government savings bonds at .8 of their face value, and gave 3/4 of them to grandchildren, nephews, and nieces for Christmas presents. If she sold the remainder later at 9/10 of their face value, what was the net amount she spent on the bonds?

A. $3350 B.$3000
C. $3450 D.$3500
E. $3250 Explanation: The net amount spent is equal to the original amount spent minus the proceeds from the sale of the remaining bonds: Original Amount = .8 × 120 × 50 = 4800 # Bonds Left after Gifts = 120 – (3/4 × 120) = 120 – 90 = 30 Sale Revenue from Leftover Bonds = 9/10 × 30 × 50 = 1350 Original Amount – Sale Revenue = 4800 – 1350 = 3450 So C is the correct answer. Fraction Problems Example 2) Mary spends her monthly take-home pay as follows: 1/3 goes to rent; 1/6 to food and clothing; 1/8 to savings; 1/4 to miscellaneous expenses. The remaining$300 she uses for entertainment. What is her monthly take-home pay?

A. $2000 B.$2500
C. $1800 D.$2100
E. $2400 Explanation: Let x be Mary’s take-home pay. Then if we take these fractions of her pay and add 300 we should get her total take-home pay: x/3 + x/6 + x/8 + x/4 + 300 = x Now we solve this equation for x. First, we note that the least common denominator for 3, 6, 8, and 4 is 24: 8x/24 + 4x/24 + 3x/24 + 6x/24 + 300 = x 21x/24 + 300 = x 21x + 7200 = 24x 7200 = 24x – 21x 7200 = 3x 2400 = x So E is the correct answer. Fraction Problems Example 3) Which of these is not equal to 0? A. 1/3 + 1/5 – 8/15 B. 2/3 + 1/15 – 11/15 C. 1/5 + 3/10 – 1/2 D. 1/2 + 1/4 – 7/8 E. 1/2 + 1/5 – 7/10 Explanation: A. 1/3 + 1/5 – 8/15 = 5/15 + 3/15 – 8/15 = 8/15 – 8/15 = 0 B. 2/3 + 1/15 – 11/15 = 10/15 + 1/15 – 11/15 = 11/15 – 11/15 = 0 C. 1/5 + 3/10 – 1/2 = 2/10 + 3/10 – 5/10 = 5/10 – 5/10 = 0 D. 1/2 + 1/4 – 7/8 = 4/8 + 2/8 – 7/8 = 6/8 – 7/8 = -1/8 E. 1/2 + 1/5 – 7/10 = 5/10 + 2/10 – 7/10 = 7/10 – 7/10 = 0 So D is the correct answer. To practice more of these types of problems, click here. Percentage Problems Example 1) Arlen gets 10% commission on his total sales above$2000. He gets no commission on sales up to $2000. If he was paid$900, what were his total sales?

A. $11,000 B.$9,000
C. $10,000 D.$12,000
E. \$11,100

Explanation: Let x be his total sales. Then

10% of (x – 2000) = 900
1/10 ∙ (x – 2000) = 900
x – 2000 = 900 ∙ 10
x = 9000 + 2000 = 11,000

So A is the correct answer.

Percentage Problems Example 2)

In a certain state between .4 percent and .8 percent of new companies will fail in their first month of business. During the current year, 8000 new companies started operations. Which of these numbers is in the range of the number of companies that failed during the first month:

A. 20
B. 45
C. 90
D. 320
E. 6400

Explanation: This problem can be a little deceptive, because it says .4 percent and .8 percent, and our natural inclination is to interpret that as 4 percent and 8 percent or as 4 tenths or 8 tenths (which would be 40% or 80%). But the actual numbers are less than 1 percent, which is less than 1/100, so we have to treat them in that way. We apply .4 percent and .8 percent therefore to the number of 8000 new companies to find out how many failed. Let x = the number of new companies that failed in their first month:

.4% of 8000 ≤ x ≤ .8% of 8000
(.4/100) ∙ 8000 ≤ x ≤ (.8/100) ∙ 8000
(4/1000) ∙ 8000 ≤ x ≤ (8/1000) ∙ 8000
4 ∙ 8 ≤ x ≤ 8 ∙ 8
32 ≤ x ≤ 64

The only number in the list of possible answers which is in this range is 45, so B is the correct answer.

Percentage Problems Example 3)

Lemon chicken is the most popular entrée at a certain Chinese restaurant. During a recent month 30% of their customers ordered take-out. If they had T customers during the month, and if 1/2 of their table service customers and 85 of their take-out customers ordered lemon chicken, how many ordered lemon chicken?

A. .50T + 85
B. .60T + 42
C. .70T + 100
D .35T + 85
E. .40T + 30

Explanation: This problem requires us to be careful in keeping concepts straight. It also presents some difficulties by using a variable, T, to represent the number of customers. It says that 30% (.3T) of their customers ordered take-out. So that means that 70% (.7T) of their customers were table service customers. If 1/2 of these ordered lemon chicken, then

1/2 of .7T = .35T, of table service customers ordered lemon chicken.