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Increases, decreases, tax, tip and interest, all as one multiplication.
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In this lesson you solve problems about percent increase and decrease, sales tax, tips, discounts, markups, simple interest and percent error. They look like many kinds of problem, but they are one: a percent change is a multiplication by a single number, so a 15% increase is multiplying by 1.15 and a 20% discount is multiplying by 0.8. Seeing them that way also shows why a 20% rise and then a 20% fall do not get you back where you started.
You know that a percent is a rate per hundred, so $35\%$ means 35 out of 100, or 0.35. You can find a percent of a number, such as $20\%$ of 60 is $0.2 \times 60 = 12$. In the last lesson you met the constant of proportionality. In this lesson you will see that every percent problem is a proportional relationship: the new amount is always a constant multiplier times the old one.
| Term | What it means |
|---|---|
| Multiplier | The single number you multiply by to make a percent change, such as 1.15 for a 15% increase or 0.8 for a 20% decrease. |
| Percent increase | The increase as a percent of the original amount. |
| Percent decrease | The decrease as a percent of the original amount. |
| Discount (markdown) | An amount taken off a price, often given as a percent of the regular price. |
| Markup | An amount a store adds to what it paid for an item, often a percent of that cost. |
| Sales tax | A percent of the price that is added on when you buy something. |
| Tip (gratuity) | Extra money given for good service, often a percent of the bill. |
| Simple interest | Money paid for the use of money: the same percent of the starting amount every year. |
| Percent error | How far an estimate or measurement is from the actual value, as a percent of the actual value. |
Every percent change can be done with one multiplication.
$$\text{new amount} = \text{multiplier} \times \text{original amount}$$
This is a proportional relationship, $y = kx$, with the multiplier as the constant $k$. Tax, tips, markups and raises are increases. Discounts, markdowns and losses are decreases. They look like different problems, but they all work the same way.
When you want the percent itself, work the other way: find the change, then divide by the original amount.
$$\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$$
Another way: picture
Draw a bar for the original price and split it into 10 equal pieces, each 10%. A 30% discount crosses out 3 pieces and leaves 7, which is 0.7 of the bar. A 30% increase adds 3 more pieces to the end, making 13 pieces, which is 1.3 of the bar.
Another way: two ways
You can find a 20% tip on a 45-dollar bill in two steps, $0.2 \times 45 = 9$ and then $45 + 9 = 54$, or in one step, $1.2 \times 45 = 54$. Both are right. The one-step way is faster and is the key to working backward.
Sales tax, tips and markups all add a percent on top of an amount.
A handy trick for tips: 10% of an amount is found by moving the decimal point one place left. For 10% of 28 dollars you get 2.80. Double it for 20% (5.60). For 15%, add half of 10% to 10%: $2.80 + 1.40 = 4.20$.
A discount of $p\%$ leaves $100\% - p\%$ of the price. A 25% discount leaves 75%, so a 48-dollar sweater costs $48 \times 0.75 = 36$ dollars.
You could also find the discount, $0.25 \times 48 = 12$ dollars, and subtract, $48 - 12 = 36$. Both ways give the same answer. The multiplier way is shorter, and it keeps you from a common slip: answering with the discount (12 dollars) instead of the sale price (36 dollars).
Sometimes you know both amounts and want the percent. Use three steps: subtract, divide by the original, multiply by 100.
A movie ticket went from 12 dollars to 15 dollars. The change is 3 dollars. Divide by the original price: $3 \div 12 = 0.25$. That is a 25% increase.
Now suppose the price later drops back from 15 dollars to 12 dollars. The change is still 3 dollars, but now the original is 15: $3 \div 15 = 0.2$, a 20% decrease. The same 3-dollar change is a different percent, because percent change always compares with the amount you started from.
When you put money in a savings account, the bank pays you interest. With simple interest, you earn the same percent of your starting amount, called the principal, every year:
$$\text{interest} = \text{principal} \times \text{rate} \times \text{years}$$
If you deposit 500 dollars at 4% simple interest, you earn $500 \times 0.04 = 20$ dollars each year. After 3 years that is $20 \times 3 = 60$ dollars of interest, and the account holds 560 dollars. Loans work the same way, except that you pay the interest.
Percent error measures how far off an estimate or a measurement is. Use the same three steps as percent change, and always divide by the actual value:
$$\text{percent error} = \frac{|\text{estimate} - \text{actual}|}{\text{actual}} \times 100\%$$
If you estimate a room is 18 feet long and it is really 20 feet, the error is 2 feet, and $2 \div 20 = 0.1$, so the percent error is 10%. The bars around the difference mean you use its size and ignore whether the estimate was high or low. An estimate of 22 feet would also be 2 feet off, so it has the same 10% error. Scientists use percent error to judge a measurement: a 2-foot error in a 20-foot room is large, but the same 2 feet in a mile-long road is tiny.
If you know the amount after a percent change, divide by the multiplier to find the original. A shirt costs 42 dollars after a 30% discount. The multiplier was 0.7, so the original price was $42 \div 0.7 = 60$ dollars. Adding 30% to 42 dollars gives 54.60 dollars, which is wrong, because the 30% was 30% of 60, not of 42.
The same idea explains why a 20% rise followed by a 20% fall does not bring you back. The two multipliers are 1.2 and 0.8, and $1.2 \times 0.8 = 0.96$. A 100-dollar price rises to 120 dollars and then falls to 96 dollars. The fall was 20% of the bigger number, so it took off more than the rise added.
Every percent problem fits one of two plans.
When you know the percent and want an amount:
When you know two amounts and want the percent:
Each step has a reason. Writing the multiplier first turns every problem into one multiplication, and dividing by the original is what makes a percent change a fair comparison.
How to check. An increase must make the amount bigger, and a discount must make it smaller. A sale price must be less than the regular price. Check a percent change by applying it to the original: it must give the new amount. And round money to the nearest cent only at the end.
Most US states charge a sales tax on things you buy, and it is added at the register rather than shown on the price tag. Five states, Alaska, Delaware, Montana, New Hampshire and Oregon, have no statewide sales tax. California has the highest statewide rate, 7.25%, and many cities add their own tax on top. A 300-dollar tablet bought where the total rate is 7.25% costs $300 \times 1.0725 = 321.75$ dollars. The same tablet in Oregon costs just 300 dollars, a difference of 21.75 dollars. For a family buying a 1,200-dollar refrigerator, the difference would be $1{,}200 \times 0.0725 = 87$ dollars, which is why some shoppers near a state line cross it for big purchases.
In the United States, it is common to leave a tip of about 15% to 20% of the bill for table service at a restaurant, because servers are often paid a low hourly wage and earn much of their pay from tips. For a 64-dollar bill, a 15% tip is $0.15 \times 64 = 9.60$ dollars and a 20% tip is $0.2 \times 64 = 12.80$ dollars. Many people use the 10% trick at the table: 10% of 64 dollars is 6.40 dollars, so 20% is double that, 12.80 dollars. The total with a 20% tip is $64 \times 1.2 = 76.80$ dollars.
Suppose a grandparent gives a 7th grader 800 dollars to save for college, and it goes into an account paying 3% simple interest per year. Each year the account earns $800 \times 0.03 = 24$ dollars. By the end of high school, 6 years later, the interest adds up to $24 \times 6 = 144$ dollars, and the account holds 944 dollars. The total grew by $144 \div 800 = 0.18$, an 18% increase, which is the 3% rate times 6 years. Loans work the other way: if you borrow 800 dollars at 3% simple interest for 6 years, you pay back 944 dollars. Comparing rates before you save or borrow is one of the most useful percent skills there is.
The most common mistake is dividing by the wrong amount. Percent change is always the change divided by the original amount, and percent error is divided by the actual amount. Dividing by the new amount gives a different, wrong percent.
The second is answering with the discount instead of the price. If a 50-dollar game is 20% off, 10 dollars is the discount; the sale price is 40 dollars.
The third is working backward by adding the percent back on. A 20% discount is undone by dividing by 0.8, not by adding 20% of the sale price.
The fourth is expecting equal percents to cancel. A 10% raise and then a 10% cut leave you with $1.1 \times 0.9 = 0.99$ of what you started with.
The last is forgetting the original in an increase: a 15% increase multiplies by 1.15, not by 0.15.
A family's dinner bill is 42.50 dollars, and they leave a 20% tip. Write the tip rate as a decimal.
$20\% = 0.2$
Percent means per hundred.
Find the tip.
$0.2 \times 42.50 = 8.50$
The tip is 20% of the bill.
Add the tip to the bill.
$42.50 + 8.50 = 51.00$
The family pays the bill and the tip together.
Check with the one-step multiplier.
$42.50 \times 1.2 = 51.00$
Adding 20% is the same as multiplying by 1.2.
Check the size with the 10% trick.
$10\% = 4.25, \quad 20\% = 8.50$
Moving the decimal point one place gives 10%, and doubling it gives 20%.
A phone's price drops from 250 dollars to 200 dollars. Find the change.
$250 - 200 = 50$
The decrease is the old price minus the new price.
Name the original amount.
$\text{original} = 250$
The price before the change is what the percent compares with.
Divide the change by the original.
$50 \div 250 = 0.2$
This gives the change as a fraction of the original price.
Write it as a percent.
$0.2 = 20\%$
Multiply by 100 to change a decimal into a percent.
Check by applying the decrease.
$250 \times 0.8 = 200$
A 20% decrease is a multiplier of 0.8, and it gives the new price.
See what dividing by the wrong price would give.
$50 \div 200 = 0.25 = 25\%$
Dividing by the new price gives a wrong answer, which is why the original must go on the bottom.
A store buys a lamp for 40 dollars and marks it up 50%. Write the markup multiplier.
$100\% + 50\% = 1.5$
A markup is an increase on what the store paid.
Find the regular price.
$40 \times 1.5 = 60$
Multiplying by 1.5 adds half of 40 to 40.
Later the lamp goes on sale at 30% off. Write the discount multiplier.
$100\% - 30\% = 0.7$
After a 30% discount, 70% of the price is left.
Find the sale price.
$60 \times 0.7 = 42$
The discount is a percent of the regular price, 60 dollars.
A shopper pays 8% sales tax. Write the tax multiplier.
$100\% + 8\% = 1.08$
Tax is added on top of the sale price.
Find what the shopper pays.
$42 \times 1.08 = 45.36$
The tax is charged on the sale price, not the regular price.
Compare the sale price with what the store paid.
$42 - 40 = 2$
Even on sale, the store still makes 2 dollars on the lamp.
Write that as a percent of the store's cost.
$2 \div 40 = 0.05 = 5\%$
A 50% markup and then a 30% discount leave the price only 5% above cost, because $1.5 \times 0.7 = 1.05$.
Maya estimated a bus trip would take 45 minutes. It took 50 minutes. Find the error.
$50 - 45 = 5$
The error is the distance between the estimate and the actual time.
Divide the error by the actual time.
Write the percent error.
Which single number do you multiply an amount by to increase it by $33\%$?
Eli guesses that a jar holds $308$ jellybeans. The actual count is $275$. Complete the worked solution to find the percent error of his guess.
Subtract the actual count from the guess.
$308 - 275 =$ d
The error is how far the guess is from the truth.
Divide the error by the actual count.
$\text{error} \div 275 =$ q
Percent error always compares the error with the actual amount.
Write the decimal as a percent.
$\text{percent error} =$ p $\%$
Multiply by 100 to change a decimal into a percent.
A bike costs $900$ dollars. Its price rises by $10\%$, and later the new price falls by $10\%$. How does the final price compare with the original?
A pair of sneakers costs $140$ dollars. The store takes $30\%$ off. What is the sale price, in dollars?
answer dollars
A theme park raised its ticket price from $180$ dollars to $252$ dollars. By what percent did the price increase?
answer%
A backpack is priced at $83$ dollars, and the sales tax rate is $7\%$. How much is the tax, and what is the total cost with tax, in dollars?
Tax: x dollars. Total: c dollars.
A skateboard is on sale for $70$ dollars, which is $30\%$ off its regular price. What is the regular price, in dollars?
answer dollars
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A jacket's regular price is $220$ dollars. It is on sale for $15\%$ off, and then $7\%$ sales tax is added to the sale price. What is the sale price, and what does the jacket cost in all, in dollars?
Sale price: s dollars. Cost in all: c dollars.
You can solve percent increase and decrease problems. Without looking: what single number do you multiply by for a 15% increase, what do you divide by to find a percent change, and why does a 20% rise followed by a 20% fall not return the original?
19. Your turn: a trip that took 50 minutes, not 45, step 2
$5 \div 50 = 0.1$
Percent error always compares with the actual value.
19. Your turn: a trip that took 50 minutes, not 45, step 3
$0.1 = 10\%$
Multiply by 100 to write a percent.