Math worksheet with 12 problems focused on solving percent problems using proportional relationships and multistep calculations.

Let's solve each problem step by step and explain the reasoning behind the solutions.

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Problem 1

In a recent town election, 1,860 people voted for either candidate A or candidate B for the position of supervisor. If candidate A received 55% of the votes, how many votes did candidate B receive?

- Total votes = 1,860
- Candidate A got 55%, so Candidate B got the remaining:
$ 100\% - 55\% = 45\% $
- Votes for B:
$ 45\% \times 1,860 = \frac{45}{100} \times 1,860 = 0.45 \times 1,860 = 837 $

✔ Answer: 2) 837

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Problem 2

Twenty-five percent of 88 is the same as what percent of 22?

- First, calculate 25% of 88:
$ 0.25 \times 88 = 22 $
- Now, we want to find what percent of 22 equals 22:
Let $ x\% $ of 22 = 22 →
$ \frac{x}{100} \times 22 = 22 $
$ \frac{x}{100} = 1 $ → $ x = 100 $

✔ Answer: 4) 100%

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Problem 3

The Edison Lightbulb Company tests 5% of their daily production of lightbulbs. If 500 bulbs were tested on Tuesday, what was the total number of bulbs produced that day?

- 5% of total = 500
$ 0.05 \times T = 500 $
$ T = \frac{500}{0.05} = 10,000 $

✔ Answer: 3) 10,000

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Problem 4

Linda paid $48 for a jacket that was on sale for 25% of the original price. What was the original price of the jacket?

- She paid 25% of the original price →
$ 0.25 \times P = 48 $
$ P = \frac{48}{0.25} = 192 $

✔ Answer: 4) $192

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Problem 5

Rashawn bought a CD that cost $18.99 and paid $20.51, including sales tax. What was the rate of the sales tax?

- Tax amount = $20.51 - $18.99 = $1.52
- Sales tax rate = $ \frac{1.52}{18.99} \times 100\% \approx 8\% $

✔ Answer: 4) 8%

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Problem 6

Carla bought a dress at a sale for 20% off the original price. The sale price of the dress was $28.80. Find the original price of the dress, in dollars.

- Sale price = 80% of original (since 20% off)
$ 0.80 \times P = 28.80 $
$ P = \frac{28.80}{0.80} = 36 $

✔ Answer: $36

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Problem 7

The Hudson Record Store is having a going-out-of-business sale. CDs normally sell for $18.00. During the first week of the sale, all CDs sell for $15.00. Written as a fraction, what is the rate of discount? What is this rate expressed as a percent? Round your answer to the nearest hundredth of a percent. During the second week of the sale, the same CDs will be on sale for 25% off the original price. What is the price of a CD during the second week of the sale?

#### Part 1: Discount from $18 to $15
- Discount amount = $18 - $15 = $3
- Fractional discount = $ \frac{3}{18} = \frac{1}{6} $
- Percent discount = $ \frac{1}{6} \times 100\% \approx 16.67\% $

#### Part 2: Second week — 25% off original price
- $ 25\% $ of $18 = $0.25 × 18 = $4.50
- Sale price = $18 - $4.50 = $13.50

✔ Answers:
- Fraction: $ \frac{1}{6} $
- Percent: 16.67%
- Second week price: $13.50

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Problem 8

A clothing store offers a 50% discount at the end of each week that an item remains unsold. Patrick wants to buy a shirt at the store and he says, “I’ve got a great idea! I’ll wait two weeks, have 100% off, and get it for free!” Explain to your friend Patrick why he is incorrect and find the correct percent of discount on the original price of a shirt.

- Each week: 50% off the current price (not cumulative)
- After Week 1: Price becomes $ 50\% $ of original → $ 0.5P $
- After Week 2: 50% off new price: $ 0.5 \times 0.5P = 0.25P $
- So, final price is 25% of original → 75% discount, not 100%

✘ Patrick is wrong because discounts are applied successively to the reduced price, not added together.

✔ Correct discount: 75%

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Problem 9

A painting that regularly sells for a price of $55 is on sale for 20% off. The sales tax on the painting is 7%. Will the final total cost of the painting differ depending on whether the salesperson deducts the discount before adding the sales tax or takes the discount after computing the sum of the original price and the sales tax on $55?

Let’s compare both scenarios:

#### Scenario 1: Discount first, then tax
- Sale price: $ 55 \times (1 - 0.20) = 55 \times 0.80 = 44 $
- Add 7% tax: $ 44 \times 1.07 = 47.08 $

#### Scenario 2: Tax first, then discount
- Add 7% tax to $55: $ 55 \times 1.07 = 58.85 $
- Apply 20% discount: $ 58.85 \times 0.80 = 47.08 $

✔ Same result: $47.08

So, no, the final cost does not differ.

> Explanation: Mathematically:
> $$
> (P \times (1 - d)) \times (1 + t) = P \times (1 + t) \times (1 - d)
> $$
> Since multiplication is commutative, order doesn’t matter.

✔ Answer: No, the final cost is the same.

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Problem 10

Shana wants to buy a new bicycle that has a retail price of $259.99. She knows that it will be on sale next week for 30% off the retail price. If the tax rate is 7%, find the total amount, to the nearest cent, that she will save by waiting until next week.

- Current price with tax:
$ 259.99 \times 1.07 = 277.19 $

- Sale price (before tax):
$ 259.99 \times (1 - 0.30) = 259.99 \times 0.70 = 181.993 $

- Sale price with tax:
$ 181.993 \times 1.07 = 194.73251 \approx 194.73 $

- Savings:
$ 277.19 - 194.73 = 82.46 $

✔ Answer: $82.46 saved

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Problem 11

Sue bought a picnic table on sale for 50% off the original price. The store charged her 10% tax and her final cost was $22.00. What was the original price of the picnic table?

Let original price = $ P $

- Sale price: $ 0.5P $
- After 10% tax: $ 0.5P \times 1.10 = 0.55P $
- Final cost: $ 0.55P = 22 $
- $ P = \frac{22}{0.55} = 40 $

✔ Answer: Original price = $40

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Problem 12

Miller's Department Store is having a sale with a 25% discount on mattresses. If the sales tax rate is 8%, how much change will Frank receive if he purchases a mattress regularly priced at $800 if he pays with $895 during this sale?

- Sale price: $ 800 \times (1 - 0.25) = 800 \times 0.75 = 600 $
- Add 8% tax: $ 600 \times 1.08 = 648 $
- He pays $895 → Change = $ 895 - 648 = 247 $

✔ Answer: $247 change

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | 837 |
| 2 | 100% |
| 3 | 10,000 |
| 4 | $192 |
| 5 | 8% |
| 6 | $36 |
| 7 | Fraction: $ \frac{1}{6} $, Percent: 16.67%, Price: $13.50 |
| 8 | Incorrect; only 75% off after two weeks |
| 9 | No, same final cost ($47.08) |
| 10 | $82.46 saved |
| 11 | $40 |
| 12 | $247 change |

Let me know if you'd like these formatted into a printable answer key!