6.2.6: Solving Problems about Percent Increase or Decrease

Lesson

Let's use tape diagrams, equations, and reasoning to solve problems with negatives and percents.

Exercise \(\PageIndex{1}\): 20% Off

An item costs \(x\) dollars and then a 20% discount is applied. Select all the expressions that could represent the price of the item after the discount.

1. \(\frac{20}{100}x\)
2. \(x-\frac{20}{100}x\)
3. \((1-0.20)x\)
4. \(\frac{100-20}{100}x\)
5. \(0.80x\)
6. \((100-20)x\)

Exercise \(\PageIndex{2}\): Walking More Each Day

1. Mai started a new exercise program. On the second day, she walked 5 minutes more than on the first day. On the third day, she increased her walking time from day 2 by 20% and walked for 42 minutes. Mai drew a diagram to show her progress.

Figure \(\PageIndex{1}\): Three tape diagrams, day 1, day 2, day 3. Diagram Day 1, 1 part, labeled d. Diagram Day 2, 2 parts, first part, d, same length as d above, second part labeled 5. Diagram day 3, has 6 equal parts with no labels. The first five parts together are the length of the Day 3 diagram above, the total is 42.

Explain how the diagram represents the situation.

2. Noah said the equation \(1.20(d+5)=42\) also represents the situation. Do you agree with Noah? Explain your reasoning.

3. Find the number of minutes Mai walked on the first day. Did you use the diagram, the equation, or another strategy? Explain or show your reasoning.

4. Mai has been walking indoors because of cold temperatures. On Day 4 at noon, Mai hears a report that the temperature is only 9 degrees Fahrenheit. She remembers the morning news reporting that the temperature had doubled since midnight and was expected to rise 15 degrees by noon. Mai is pretty sure she can draw a diagram to represent this situation but isn't sure if the equation is \(9=15+2t\) or \(2(t+15)=9\). What would you tell Mai about the diagram and the equation and how they might be useful to find the temperature, \(t\), at midnight?

Exercise \(\PageIndex{3}\): A Sale on Shoes

1. A store is having a sale where all shoes are discounted by 20%. Diego has a coupon for $3 off of the regular price for one pair of shoes. The store first applies the coupon and then takes 20% off of the reduced price. If Diego pays $18.40 for a pair of shoes, what was their original price before the sale and without the coupon?
2. Before the sale, the store had 100 pairs of flip flops in stock. After selling some, they notice that \(\frac{3}{5}\) of the flip flops they have left are blue. If the store has 39 pairs of blue flip flops, how many pairs of flip flops (any color) have they sold?
3. When the store had sold \(\frac{2}{9}\) of the boots that were on display, they brought out another 34 pairs from the stock room. If that gave them 174 pairs of boots out, how many pairs were on display originally?
4. On the morning of the sale, the store donated 50 pairs of shoes to a homeless shelter. Then they sold 64% of their remaining inventory during the sale. If the store had 288 pairs after the donation and the sale, how many pairs of shoes did they have at the start?

Are you ready for more?

A coffee shop offers a special: 33% extra free or 33% off the regular price. Which offer is a better deal? Explain your reasoning.

Summary

We can solve problems where there is a percent increase or decrease by using what we know about equations. For example, a camping store increases the price of a tent by 25%. A customer then uses a $10 coupon for the tent and pays $152.50. We can draw a diagram that shows first the 25% increase and then the $10 coupon.

Figure \(\PageIndex{2}\): Three tape diagrams of unequal length. Top diagram, original price, one part labeled p. Middle diagram, 25% increase, 4 equal parts which total to the same length as p above, with another equal part on the end labeled point 25 p. Third diagram, 10 dollar coupon, first part the same length as three parts above, labeled 152 point 50, second part, dotted outline, same length as two parts above, labeled 10.

The price after the 25% increase is \(p+.25p\) or \(1.25p\). An equation that represents the situation could be \(1.25p-10=152.50\). To find the original price before the increase and discount, we can add 10 to each side and divide each side by 1.25, resulting in \(p=130\). The original price of the tent was $130.

Practice

Exercise \(\PageIndex{4}\)

A backpack normally costs $25 but it is on sale for $21. What percentage is the discount?

(From Unit 4.3.3)

Exercise \(\PageIndex{5}\)

Find each product

1. \(\frac{2}{5}\cdot (-10)\)
2. \(-8\cdot\left(\frac{-3}{2}\right)\)
3. \(\frac{10}{6}\cdot 0.6\)
4. \(\left(\frac{-100}{37}\right)\cdot\left(-0.37\right)\)

(From Unit 5.3.2)

Exercise \(\PageIndex{6}\)

Select all expressions that show \(x\) increased by 35%.

1. \(1.35x\)
2. \(\frac{35}{100}x\)
3. \(x+\frac{35}{100}x\)
4. \((1+0.35)x\)
5. \(\frac{100+35}{100}x\)
6. \((100+35)x\)

Exercise \(\PageIndex{7}\)

Complete each sentence with the word discount, deposit, or withdrawal.

1. Clare took $20 out of her bank account. She made a _____.
2. Kiran used a coupon when he bought a pair of shoes. He got a _____.
3. Priya put $20 into her bank account. She made a _____.
4. Lin paid less than usual for a pack of gum because it was on sale. She got a _____.

(From Unit 4.3.2)

Exercise \(\PageIndex{8}\)

Here are two stories:

• The initial freshman class at a college is 10% smaller than last year’s class. But then during the first week of classes, 20 more students enroll. There are then 830 students in the freshman class.
• A store reduces the price of a computer by $20. Then during a 10% off sale, a customer pays $830.

Here are two equations:

• \(0.9x+20=830\)
• \(0.9(x-20)=830\)

1. Decide which equation represents each story.
2. Explain why one equation has parentheses and the other doesn’t.
3. Solve each equation, and explain what the solution means in the situation.