# The Craft of Word Problems

## CONTENTS OF CURRICULUM UNIT 04.05.04

## The Art of Interpreting Percent

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## Lesson 5

Goal: The students should have an understanding of the relationship between fractions and percentages. They should also be comfortable converting fractions to decimals and percentages and vice versa. At this point in the unit the students will be exposed to formulas to solve real world problems. When solving the word problems in this unit the following notations will be utilized.

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Notation

- Q = Whole Quantity
- P = Percent
- I = Incremental Part of the Whole Quantity
- D = Decimal
- T = Total

### Decimal conversions

I will explain that many calculators do not have the “percent key” and that students should possess the knowledge to calculate percentage problems utilizing a decimal. Therefore, the earlier lessons of changing a percent to a decimal will be instrumental in order to solve the following problems. The corresponding formulas are listed below for conversions of Decimal to Percent and vice versa.

The Decimal is calculated by dividing the percent, without the% sign, by 100.

Decimal = P/100

0.75 = 75 / 100

Decimal = P/100

1.25 = 125/100

The Percent is calculated by multiplying the Decimal by 100, then adding a% sign to the answer. Listed below are the calculations.

Percent = D x 100

75% = 0.75 x 100

Percent = D x 100

125% = 1.25 x 100

Revisit the principal that Percent = Fraction=Decimal therefore, 75% = 75/100 = 0.75

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### Sales Tax

Students will appreciate how to determine the sales tax as a function of the cost of an item. In Connecticut the sales tax on non essential purchases is 6% and students should understand how the tax is calculated. The following lesson teaches the practical, real world aspects of calculating percent. When one encounters sales tax it is usually listed as a Percent of the Whole Quantity and therefore, added to the Total Cost.

### Solving for the Incremental Part of the Whole “I” when the Percent “P” and the Whole Quantity “Q” are known:

The most common type of percentage problem is one in which the Percent and the Whole Quantity are known and the Incremental Part of the Whole must be calculated.

Solving for the Incremental Part of the Whole using Percent:

Problem 1. Tyesha purchased a CD for $20 and the sales tax was 6%, how much did Tyesha pay for the CD including the sales tax?

Whole Quantity Q x Percent P = Incremental Part of the Whole Quantity I

$20 x 6% = $1.20

Whole Quantity Q + Incremental Part of the Whole Quantity I = Total T

$20 + $1.20 = $21.20

Answer 1. Tyesha paid $21.20 for the CD including the sales tax.

Solving for the Incremental Part of the Whole using a Decimal:

Problem 2. Tyesha purchased a CD for $20 and the sales tax was 6%, how much did Tyesha pay for the CD including the sales tax?

Whole Quantity Q x Decimal D = Incremental Part of the Whole Quantity I

$20 x 0.06 = $1.20

Whole Quantity Q + Incremental Part of the Whole Quantity I = Total T

$20 + $1.20 = $21.20

Answer2. Tyesha paid $21.20 for the CD including the sales tax.

Solving for the Incremental Part of the Whole Quantity, using Percent:

Problem 3. Hercules purchased a basketball for $18.50, the sales tax was 6%. How much did Hercules pay for the basketball including the sales tax?

Q x P = I

$18.50 x 6% = $1.11

Q + I = T

$18.50 + $1.11 = $19.61

Answer 3. Hercules spent $19.61 for the basketball including the sales tax.

Solving for the Incremental Part of the Whole, using a Decimal:

Problem 4. Hercules purchased a basketball for $18.50, the sales tax was 6%. How much did Hercules pay for the basketball including the sales tax?

Q x D = I

$18.50 x .06 = $1.11

Q + I = T

$18.50 + $1.11 = $19.61

Answer 4. Hercules spent $19.61 for the basketball including the sales tax.

### Solving for Percent “P” when the Whole Quantity “Q” and the Incremental Part of the Whole Quantity “I” are Known.

One may encounter problems where knowledge of the Whole Quantity Q and Incremental Part of the Whole Quantity I are given and the Percent P must be determined. In this case the formula is listed below:

Decimal D = Incremental Part of the Whole Quantity I / Whole Quantity Q D = I / Q

Percent P = Decimal D x 100

Problem 5. Assume that Juanita purchased a baseball cap for $15.00 and the sales tax was $0.90, what percentage was the sales tax.

D = I / Q

D = $0.90 / $15.00

.06 = $0.90 / $15.00

D = .06

P = D x 100

P = 0.06 x 100 = 6%

Answer 5. Juanita was charged 6% sales tax on the purchase of a baseball cap.

Problem 6. Miguel mows lawns in the neighborhood and earns $25.00 per week. He gives $2.00 to his church each week, what percentage of his earnings does he contribute to his church weekly?

D = I / Q

D = $2.00 / $25.00

D = 0.08

P = D x 100

P = 0.08 x 100 = 8%

Answer 6. Miguel contributes 8% of his earnings to church each week.

### Solving for the Whole Quantity “Q” when the Percent “P” and the Incremental Part of the Whole Quantity “I” are Known:

A third situation occurs when the Percent and Incremental Part of the Whole Quantity are known and the Whole Quantity must be calculated. In this case the formula is listed below.

Solving for the Whole Quantity utilizing percent:

Problem 7. Carlos ate lunch at his favorite restaurant and left a $12.00 tip. The tip represents 20% of the cost of the meal, how much did the meal cost?

Q = I / P

Q = $12.00 / 20%

Q = $60.00

Answer 7. Carlos spent $60.00 on his lunch without the tip. The entire cost for the lunch including the tip was $60.00 + $12.00 = $72.00

Solving for the Whole Quantity utilizing a decimal:

Problem 8. Carlos ate lunch at his favorite restaurant and left a $12.00 tip. The

tip represents 20% of the cost of meal, how much did the meal cost?

Q = I / D

Q = $12.00 / 0.20

Q = $60.00

Answer 8. Carlos spent $60.00 on his lunch without the tip. The entire cost for the lunch including the tip was $60.00 + $12.00 = $72.00

Solving for the Whole Quantity utilizing percent:

Problem 9. Janaya is a waitress at the Rusty Scupper. She earns 15% of the dinner bill in tips for parties of eight and over. Saturday night she made $37.50 in tips for a party of ten. How much was the dinner bill? How much did the party of ten pay for dinner including the tip?

Q = I / P

Q = $37.50 / 15%

Q = $250.00

Answer 9. Janaya’s party of ten spent $250 for dinner. The total bill including the tip was $250 + $37.50 = $287.50.

Solving for the Whole Quantity utilizing a decimal

Problem 10. Janaya is a waitress at the Rusty Scupper. She earns 15% of the dinner bill in tips for parties of eight and over. Saturday night she made $37.50 in tips for a party of ten. How much was the dinner bill? How much did the party of ten pay for dinner including the tip?

Q = I / D

Q = $37.50 / 0.15

Q = $250.00

Answer 10. Janaya’s party of ten spent $250 for dinner. The total bill including the tip was $250 + $37.50 = $287.50.

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