Solving Percent Problems Study Guide

Introduction to Solving Percent Problems

The hardest thing in the world to understand is the income tax.

—ALBERT EINSTEIN, theoretical physicist (1879–1955)

The final percent lesson focuses on another approach to solving percent problems, one that is more direct than the approach described in the previous lesson. It also gives some shortcuts for ﬁnding particular percents and teaches how to calculate percent of change (the percent that a ﬁgure increases or decreases).

There is a more direct approach to solving percent problems than the shortcut formula you learned in the previous percent lesson:

The direct approach is based on the concept of translating a word problem practically word for word from English statements into mathematical statements. The most important translation rules you'll need are:

of means multiply (×)
is means equals (=)

You can put this direct approach to work on the three main varieties of percent problems.

Example: What is 15% of 50? (50 is the whole.)

Translation:

The word What is the unknown quantity; use the variable w to stand for it.
The word is means equals (=).
Mathematically, 15% is equivalent to both 0.15 and (your choice, depending on whether you prefer to work in decimals or fractions).
of 50 means multiply by 50 (×50).

Put it all together as an equation and solve it:

	w = 0.15 × 50	OR
	w = 7.5

Thus, 7.5 (which is the same as ) is 15% of 50.

Tip

Sam's $50 video game is 20% off today. What will the sale price be? There's a shortcut to questions like this. Rather than ﬁnding 20% of $50 and then subtracting it from $50, think about it this way: if it's 20% off, then Sam will pay 80% of the game's original price. 80% of $50 is 0.80 × $50 = $40, so the sale price is $40.

Finding What Percent One Number Is of Another Number

Example: 10 is what percent of 40?

Translation:

10 is means 10 is equal to (10 =).
What percent is the unknown quantity, so let's use to stand for it. (The variable w is written as a fraction over 100 because the word percent means per 100,or over 100.)
of 40 means multiply by 40 (× 40).

Put it all together as an equation and solve:
Write 10 and 40 as fractions:
Multiply fractions:
Reduce:
Cross multiply:	10 × 5 = w × 2
Solve by dividing both sides by 2:	25 = w
Thus, 10 is 25% of 40

Tip

Caution! Since the variable w is being written above a 100 denominator, it is being written as a percent and not as a decimal. Therefore, do not move the decimal of your answer—just add the % symbol.

Finding the Whole When a Percent Is Given

Example: 20 is 40% of what number?

Translation:

20 is means 20 equal to (20 =).
Mathematically, 40% is equivalent to both 0.40 (which is the same as 0.4) and (which reduces to ). Again, it's your choice, depending on which form you prefer.
of what number means multiply by the unknown quantity; let's use w for it (× w).

Put it all together as an equation and solve:

20 = 0.4 ×w	OR	20 =


20 ÷ 0.4 = w ÷ 0.4		20 × 5 =	2 × w
		100 =	2 × w
50 = w		100 =	2 × 50
Thus, 20 is 40% of 50.

The 15% Tip Shortcut

Have you ever been in the position of getting your bill in a restaurant and not being able to quickly calculate an appropriate tip (without using a calculator or giving the bill to a friend)? If that's you, read on.

It's actually faster to calculate two ﬁgures—10% of the bill and 5% of the bill—and then add them together.

Calculate 10% of the bill by moving the decimal point one digit to the left.
Examples:
- 10% of $35.00 is $3.50.
- 10% of $82.50 is $8.25.
- 10% of $59.23 is $5.923, which rounds to $5.92.
Pretty easy, isn't it?
Calculate 5% by taking half of the amount you calculated in step 1.

Examples:
- 5% of $35.00 is half of $3.50, which is $1.75.
- 5% of $82.50 is half of $8.25, which is $4.125, which rounds to $4.13.
- 5% of $59.23 is approximately half of $5.92, which is $2.96. (We said approximately because we rounded $5.923 down to $5.92.We're going to be off by a fraction of a cent, but that really doesn't matter—you're probably going to round the tip to a more convenient amount, like the nearest nickel or quarter.)
Calculate 15% by adding together the results of step 1 and step 2.

Examples:
- 15% of $35.00 = $3.50 + $1.75 = $5.25.
- 15% of $82.50 = $8.25 + $4.13 = $12.38
- 15% of $59.23 = $5.92 + $2.96 = $8.88

You might want round each calculation up to a more convenient amount of money to leave, such as $5.50, $12.50, and $9 if your server was good; or round down if your service wasn't terriﬁc.

Percent of Change (% Increase and % Decrease)

You can use the technique to ﬁnd the percent of change, whether it's an increase or a decrease. The is number is the amount of the increase or decrease, and the of number is the original amount.

Example: If a merchant puts his $10 pens on sale for $8, by what percent does he decrease the selling price?

1. Calculate the decrease, the is number:	$10 – $8 = $2
2. The of number is the original amount:	$10
3. Set up the formula and solve for the % by cross multiplying:
	2 × 100 = 10 × %
	200 = 10 × %
	200 = 10 × 20

Thus, the selling price is decreased by 20%.

If the merchant later raises the price of the pens from $8 back to $10, don't be fooled into thinking that the percent increase is also 20%! It's actually more, because the increase amount of $2 is now based on a lower original price of only $8 (since he's now starting from $8):
		2 × 100 = 8 × %
		200 = 8 × %
		200 = 8 × 25
Thus, the selling price is increased by 25%.

Alternatively, you can use a more direct approach to finding the percent of change by setting up the following formula:

Here's the solution to the previous questions using this more direct approach: Price decrease from $10 to $8:

1.	Calculate the decrease:	$10 – $8 = $2
2.	Divide it by the original amount, $10, and multiply by 100 to change the fraction to a percent:

Thus, the selling price is decreased by 20%. Price increase from $8 back to $10:

1.	Calculate the decrease:	$10 – $8 = $2
2.	Divide it by the original amount, $10, and multiply by 100 to change the fraction to a percent:

Thus, the selling price is increased by 25%.

Tip

Here's a shortcut to finding the total price of an item with tax. Write the tax as a decimal, and then add it to one (8% tax would become 1.08). Then, multiply that by the cost of the item. The product is the total price. Example: What would a $34 vase cost if tax is 6.5%? $34 × 1.065 = $36.21, which would be the final cost.

Tip

The next time you eat in a restaurant, figure out how much of a tip to leave your server without using a calculator. In fact, figure out how much 15% of the bill is and how much 20% of the bill is, so you can decide how much tip to leave. Perhaps your server was a little better than average, so you want to leave a tip slightly higher than 15%, but not as much as 20%. If that's the case, figure out how much money you should leave as a tip. Do you remember the shortcut for figuring tips from this lesson?

Solving Percent Problems Sample Questions

Ninety percent of the 300 dentists surveyed recommended sugarless gum for their patients who chew gum. How many dentists did NOT recommend sugarless gum?
The quality control step at the Light Bright Company has found that 2 out of every 1,000 light bulbs tested are defective. Assuming that this batch is indicative of all the light bulbs they manufacture, what percent of the manufactured light bulbs is defective?
The combined city and state sales tax in Bay City is %. The Bay City Boutique collected $600 in sales tax on May 1. What was the total sale ﬁgure for that day, excluding sales tax?
If your server was especially good or you ate at an expensive restaurant, you might want to leave a 20% tip. Can you ﬁgure out how to quickly calculate it?

Solutions to Sample Questions

Question 1

Translate:

90%is equivalent to both 0.9 and
of the 300 dentists means × 300
How many dentists is the unknown quantity: We'll use d for it.

But, wait! Ninety percent of the dentists did recommend sugarless gum, but we're asked to find the number of dentists who did NOT recommend it. So there will be an extra step along the way. You could find out how many dentists did recommend sugarless gum, and then subtract from the total number of dentists to find out how many did not. But there's an easier way:

Subtract 90% (the percent of dentists who did recommend sugarless gum) from 100% (the percent of dentists surveyed) to get 10% (the percent of dentists who did NOT recommend sugarless gum).

There's one more translation before you can continue: 10% is equivalent to both 0.10 (which is the same as 0.1) and (which reduces to ).

0.1 × 300 = d	OR		d
30 = d			d

Thus, 30 dentists did NOT recommend sugarless gum.

Question 2

Although you have learned that of means multiply, there is an exception to the rule. The words out of mean divide; specifically, 2 out of 1,000 light bulbs means of the light bulbs are defective. We can equate (=) the fraction of the defective light bulbs to the unknown percent that is defective, or . (Remember, a percent is a number divided by 100.) The resulting equation and its solution are shown here.

Translate:
Cross multiply:	2 × 100 =	1,000 × d
	200 =	1,000 × d
Solve for d:	200 =	1,000 × 0.2
Thus, 0.2% of the light bulbs are assumed to be defective.

Question 3

Translate:

Tax = , which is equivalent to both and 0.085
Tax =$600
Sales is the unknown amount; we'll use S to represent it.
Tax = of sales (× S)

Fraction approach:

Translate:
Rewrite 600 and S as fractions:
Multiply fractions:
Cross multiply:	600 × 100 =	1 × ×S
Solve for S by dividing both sides of the equation by :	60,000 =	× S

	7,058.82 ≈	S

Decimal approach:

Translate and solve for S by dividing by 0.085:	600 =	0.085 × S
	600 ÷ 0.085 =	0.085 × S ÷ 0.085
Rounded to the nearest cent and excluding tax,	7,058.82 ≈	S
$7,058.82 is the amount of sales on May 1.

Question 4

To quickly calculate a 20% tip, find 10% by moving the decimal point one digit to the left, and then double that number.

Find practice problems and solutions for these concepts at Solving Percent Problems Practice Questions.

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Solving Percent Problems Study Guide

Introduction to Solving Percent Problems

Tip

Finding What Percent One Number Is of Another Number

Tip

Finding the Whole When a Percent Is Given

The 15% Tip Shortcut

Percent of Change (% Increase and % Decrease)

Tip

Tip