Free ACT Math Integrating Essential Skills

ACT Math · Integrating Essential Skills · free lesson

Percentages

The idea, then worked examples solved step by step, then practice questions whose numbers change every time you try them.

The topic in one minute

Press Play: the idea and a worked example, explained aloud (about 1:23).

Percentages

Percent means out of 100.

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Read what is said

A percent is a number of hundredths, so thirty-five percent means thirty-five out of one hundred. Percent questions on the ACT are usually short stories with several steps, so it pays to know the rules well.
To find a percent of an amount, write the percent as a decimal and multiply. The whole is always the number after the word of.
Here is a sweater priced at forty-eight dollars. It is on sale for twenty-five percent off. Taking off twenty-five percent means keeping seventy-five percent, so we multiply the price by that share.
Next, a sales tax of seven and a half percent is added to the sale price, not to the original price. So we keep one hundred seven and a half percent of thirty-six.
Now the trap. The thirty-six is only the first step, and it is a wrong option on the test. Adding the tax to the full price gives fifty-one dollars and sixty cents, and one combined change of seventeen and a half percent gives thirty-nine dollars and sixty cents. Both are wrong.
Percents taken one after another multiply, they do not add. Take twenty percent off a television and then fifteen percent off the reduced price. The multipliers eighty percent and eighty-five percent multiply to give the final share of the price.
Remember this: every percent is a percent of something. Find that base before you calculate. In a change, the base is the amount before the change. Now practice it with new numbers.

The idea

Percent means "out of 100"

A percent is a number of hundredths. So 35% is $\frac{35}{100}$ , the decimal 0.35. Watch the grid of 100 squares fill: each square is 1% of the whole, and 35 shaded squares are 35% of it.

Percent of a number, and what percent

To find a percent of an amount, multiply the amount by the percent written as a decimal. To find what percent a part is of a whole, divide the part by the whole and multiply by 100. The whole is the number after the word "of".

\begin{aligned} part &= \frac{35}{100} \times 80 \\ &= 28 \\ p &= \frac{100 \times 18}{45} \\ &= 40 \end{aligned}

So 35% of 80 is $28$ , and 18 is $40$ % of 45. A part can be bigger than its whole: 90 is 150% of 60.

A percent change is one multiplication

To raise an amount by 20%, keep all of it (100%) and add 20% more: that is 120% of it, so multiply by 1.2. To lower it by 20%, keep 80%: multiply by 0.8. The number you multiply by is the multiplier. Drag the slider and watch a bar of 80 grow or shrink by the percent you choose.

The percent change goes the other way: the change divided by the amount before the change, times 100. A price that goes from 40 dollars to 52 dollars has risen by

\begin{aligned} change &= 52 - 40 \\ &= 12 \\ r &= \frac{100change}{40} \\ &= \frac{100 \times 12}{40} \\ &= 30 \end{aligned}

that is, $30$ %. Dividing by the new price, 52, is the usual slip.

Changes in a row multiply

A price of 100 dollars rises by 20%, and the new price then falls by 20%. The fall is taken on 120, not on 100, so the price does not come back.

\begin{aligned} a &= 1.2 \times 100 \\ &= 120 \\ b &= 0.8a \\ &= 0.8 \times 120 \\ &= 96 \\ k &= 1.2 \times 0.8 \\ &= 0.96 \end{aligned}

The price ends at $96$ dollars. The multipliers 1.2 and 0.8 multiply to $0.96$ : a loss of $4$ %, not 0%.

Warning.

Percents taken one after another do not add. Two discounts of 10% and 20% leave 0.9 × 0.8 of the price, which is a discount of $28$ %, not 30%. On the ACT the added percent is always among the options.

Back to the original

After a 25% increase a ticket costs 90 dollars. The 90 is 1.25 times the old price, so divide by 1.25. Taking 25% off the 90 gives a wrong answer, because that 25% would be of the new price.

\begin{aligned} (1 + \frac{25}{100})x &= 90 \\ x &= 72 \\ w &= (1 - \frac{25}{100}) \times 90 \\ &= 67.5 \end{aligned}

The ticket cost $72$ dollars. The wrong way gives $67.5$ , and raising $67.5$ by 25% does not give 90.

Markup, discount, tax and tip

Store prices are chains of percents. A store buys a jacket for 60 dollars and marks it up 50%. Later it is sold at 20% off, and a sales tax of 6% is added at the register. Each step multiplies the amount from the step before:

\begin{aligned} price &= (1 + \frac{50}{100}) \times 60 \\ &= 90 \\ sale &= (1 - \frac{20}{100})price \\ &= (1 - \frac{20}{100}) \times 90 \\ &= 72 \\ paid &= (1 + \frac{6}{100})sale \\ &= (1 + \frac{6}{100}) \times 72 \\ &= 76.32 \end{aligned}

The customer pays $76.32$ dollars. Every amount on the way, $90$ and $72$ , is a wrong option on a test question. A tip is a percent of the bill before tax unless the question says otherwise.

Simple interest

With simple interest, an amount $P$ (the principal) earns the same interest every year: $I = Prt$ , where $r$ is the yearly rate as a decimal and $t$ is the time in years. For 2,000 dollars at 4% a year for 6 years:

\begin{aligned} I &= 2000 \times \frac{4}{100} \times 6 \\ &= 480 \\ B &= 2000 + I \\ &= 2000 + 480 \\ &= 2480 \end{aligned}

The interest is $480$ dollars and the balance is $2480$ dollars. Because the interest is the same each year, the balance grows along a straight line. Watch it draw itself in.

A percent of a percent

In a school, 40% of the students are in clubs, and 25% of the students in clubs are in the chess club. The chess club is 25% of 40% of the school. Multiply the decimals:

\begin{aligned} s &= \frac{25}{100} \times \frac{40}{100} \\ &= 0.1 \end{aligned}

That is $0.1$ of the school, or $10$ %. In a school of 1,200 students, it is $120$ students.

Rule.

Every percent is a percent of something. Before you calculate, find that base: the number after "of" or "than", and in a change, the amount before the change.

Note.

On the ACT this topic comes as: a percent of an amount; what percent one number is of another; a sale price with tax; a percent increase or decrease; changes in a row; the price before a change; markup and tip; simple interest; and percents compared in a table. Most are stories in several steps. The wrong options are the numbers you meet on the way, the percents added instead of multiplied, and the change divided by the wrong base.

Worked examples

Each example is one of the kinds of question the ACT asks about percentages, from a sale price with tax to a percent of a percent. Most take several steps, and the wrong options on the test are the numbers met on the way. The numbers in the working are found by the engine.

Example.

1. A sale price with tax. A sweater is priced at 48 dollars. It is on sale for 25% off, and a sales tax of 7.5% is added to the sale price. What does the customer pay?

Two changes in a row, each one a multiplication of the amount before it. Keeping 75% of the price takes the 25% off; 107.5% of the sale price adds the tax.

\begin{aligned} s &= (1 - \frac{25}{100})P \\ &= (1 - \frac{25}{100}) \times 48 \\ &= 36 \\ T &= (1 + \frac{7.5}{100})s \\ &= (1 + \frac{7.5}{100}) \times 36 \\ &= 38.7 \end{aligned}

The customer pays $38.70$ dollars. The sale price $36$ is only the first step. Adding the tax to the full price, without the sale, would give $51.60$ dollars, and taking 25% off and adding 7.5% as one change of 17.5% off gives $39.60$ : neither is right.

Example.

2. Percent change from a table. The table gives the number of members of three clubs at a school in two years. Which club had the greatest percent increase?

Club	Last year	This year
Chess	$40$	$52$
Robotics	$25$	$35$
Drama	$64$	$80$

For each club, divide the increase by last year's number, the amount before the change, and multiply by 100. The engine works all three at once: $d$ holds the increases and $p$ the percent increases, in the order of the table.

\begin{aligned} d &= b - a \\ &= [12, 10, 16] \\ p &= \frac{100d}{a} \\ &= [30, 40, 25] \end{aligned}

Robotics rose by $40$ %, more than chess ( $30$ %) and drama ( $25$ %). Drama had the largest increase in members, $16$ , but the smallest percent increase, because it started bigger. The ACT asks both ways: read whether the question wants the change or the percent change.

Example.

3. Two discounts in a row. A store takes 20% off a television, and then a coupon takes 15% off the reduced price. What single discount would give the same final price?

Each discount keeps a share of the price before it: 80%, then 85%. The final price is the product of the two multipliers times the original price.

\begin{aligned} k &= (1 - \frac{20}{100}) \times (1 - \frac{15}{100}) \\ &= 0.68 \\ D &= 100(1 - k) \\ &= 100 \times (1 - 0.68) \\ &= 32 \end{aligned}

The final price is $0.68$ of the original, so the single discount is $32$ %, not 35%. The second discount is taken on a smaller amount, so it takes off less than 15% of the original.

Example.

4. The price before the sale and the tax. A customer paid 72.08 dollars for a pair of shoes. The price had been reduced by 15%, and a sales tax of 6% was then added. What was the price before the sale?

Work backward. The 72.08 is 1.06 times the sale price, and the sale price is 0.85 times the original price $x$ . Undo both multiplications by solving one equation.

\begin{aligned} (1 - \frac{15}{100}) \times (1 + \frac{6}{100})x &= paid \\ (1 - \frac{15}{100}) \times (1 + \frac{6}{100})x &= 72.08 \\ x &= 80 \\ s &= (1 - \frac{15}{100})x \\ &= (1 - \frac{15}{100}) \times 80 \\ &= 68 \\ check &= (1 + \frac{6}{100})s \\ &= (1 + \frac{6}{100}) \times 68 \\ &= 72.08 \end{aligned}

The shoes cost $80$ dollars before the sale. The check runs the chain forward: a sale price of $68$ dollars and $72.08$ dollars with tax. Taking 15% and 6% off 72.08 instead would treat those percents as percents of the amount paid, which they are not.

Example.

5. A restaurant bill with tax and tip. A meal costs 64 dollars before tax. The tax is 8% of the cost of the meal, and the diners leave a tip of 18% of the cost of the meal before tax. What is the total they pay?

Both percents are taken of the same base, the 64 dollars, so here they may be added. That is different from changes in a row.

\begin{aligned} tax &= \frac{8}{100}M \\ &= \frac{8}{100} \times 64 \\ &= 5.12 \\ tip &= \frac{18}{100}M \\ &= \frac{18}{100} \times 64 \\ &= 11.52 \\ total &= M + tax + tip \\ &= 64 + 5.12 + 11.52 \\ &= 80.64 \end{aligned}

The total is $80.64$ dollars: tax $5.12$ and tip $11.52$ . A tip of 18% of the bill with tax would be $12.44$ dollars, a little more; read which base the question names.

Example.

6. Simple interest. Rosa puts 3,500 dollars in an account that pays simple interest at 3.2% a year. After how many years will she have earned 560 dollars in interest?

Simple interest is $I = Prt$ . The interest, the principal and the rate are known; solve for the time $t$ in years.

\begin{aligned} y &= \frac{3.2}{100}P \\ &= \frac{3.2}{100} \times 3500 \\ &= 112 \\ P\frac{3.2}{100}t &= I \\ 3500 \times \frac{3.2}{100}t &= 560 \\ t &= 5 \end{aligned}

She earns $112$ dollars each year, so it takes $5$ years. Using 3.2 instead of 0.032 for the rate gives a time 100 times too small.

Example.

7. A percent of a percent. In a survey of 500 voters, 60% favor a new park. Of the voters who favor it, 35% say they favor it strongly. How many of the voters surveyed favor the park strongly?

The 35% is a percent of the voters who favor the park, not of all 500. Multiply the two percents as decimals, then take that share of 500.

\begin{aligned} f &= \frac{60}{100}N \\ &= \frac{60}{100} \times 500 \\ &= 300 \\ s &= \frac{35}{100}f \\ &= \frac{35}{100} \times 300 \\ &= 105 \\ share &= \frac{35}{100} \times \frac{60}{100} \\ &= 0.21 \end{aligned}

$105$ voters favor the park strongly, which is $21$ % of all the voters. The picture shows it on 100 squares: 60 are shaded for "favor", and 35% of those 60, which is $21$ squares, are dark. Taking 35% of all 500 gives $175$ , the usual wrong answer.

Practice

Answer ACT-style questions on percentages. Each answer is checked, with the full solution, and every try brings new numbers. Practice needs a free account: it keeps your progress.

Practise percentages with new numbers