How to solve percent problems

How to solve percent problems?

Read,3 minutes

Most percent problems use three quantities: the part, the whole, and the percent. The percent should be changed to a decimal before it is used in an equation.

The Percent Equation

The key relationship is $\text{part} = \text{percent} \times \text{whole}$. If one value is missing, replace it with a variable and solve.

Three Common Question Types

To find the part, multiply: $\text{part} = p \times \text{whole}$.
To find the percent, divide and multiply by $100$: $p = \frac{\text{part}}{\text{whole}} \times 100$.
To find the whole, divide: $\text{whole} = \text{part} \div p$, where $p$ is written as a decimal.

For example, if $18$ is $30\%$ of a number, write $0.30x=18$. Dividing by $0.30$ gives $x=60$.

Percent Problems

Think of this lesson as more than a rule to memorize. Percent Problems is about parts, wholes, rates, discounts, tax, and percent change. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Percent means ?per 100,? so $p\%$ is $\frac{p}{100}$. Most percent problems come from part = percent $\times$ whole.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Percent Problems

1) What number is $20\%$ of $45$?

2) $12$ is what percent of $40$?

3) $18$ is $30\%$ of what number?

4) A team won $15$ of $20$ games. What percent did it win?

5) Find $6\%$ of $250$.

6) $54$ is $75\%$ of what number?

7) What number is $12.5\%$ of $96$?

8) $8.4$ is what percent of $56$?

9) A store sold $36$ of $48$ tickets. What percent were sold?

10) $140\%$ of what number is $84$?

11) What number is $0.4\%$ of $3,500$?

12) $63$ is what percent of $180$?

13) A recipe uses $3$ cups from a $12$-cup bag. What percent of the bag is used?

14) $22.5$ is $15\%$ of what number?

15) What number is $\frac{3}{8}$ of $64$? Give the answer and the percent used.

16) $5$ is what percent of $8$?

17) A score rose from $72$ possible points to $90$ possible points, and a student earned $81$. What percent of $90$ is $81$?

18) $2.7$ is $4.5\%$ of what number?

19) After spending $35\%$ of her money, Mia spent $\$28$. How much did she start with?

20) $156$ is $130\%$ of what number?

Answers

1) $20\%=0.20$. Then $0.20 \times 45=9$. Answer: $9$.

2) $\frac{12}{40} \times 100=30$. Answer: $30\%$.

3) Let the number be $x$. $0.30x=18$, so $x=18 \div 0.30=60$.

4) $\frac{15}{20} \times 100=75$. Answer: $75\%$.

5) $6\%=0.06$. Then $0.06 \times 250=15$. Answer: $15$.

6) Let the number be $x$. $0.75x=54$, so $x=54 \div 0.75=72$.

7) $12.5\%=0.125$. Then $0.125 \times 96=12$. Answer: $12$.

8) $\frac{8.4}{56} \times 100=0.15 \times 100=15$. Answer: $15\%$.

9) $\frac{36}{48} \times 100=0.75 \times 100=75$. Answer: $75\%$.

10) Let the number be $x$. $1.40x=84$, so $x=84 \div 1.40=60$.

11) $0.4\%=0.004$. Then $0.004 \times 3,500=14$. Answer: $14$.

12) $\frac{63}{180} \times 100=0.35 \times 100=35$. Answer: $35\%$.

13) $\frac{3}{12} \times 100=25$. Answer: $25\%$.

14) Let the number be $x$. $0.15x=22.5$, so $x=22.5 \div 0.15=150$.

15) $\frac{3}{8}=37.5\%$. Then $\frac{3}{8} \times 64=24$. Answer: $24$ using $37.5\%$.

16) $\frac{5}{8} \times 100=62.5$. Answer: $62.5\%$.

17) Use $\frac{81}{90} \times 100$. This equals $0.9 \times 100=90$. Answer: $90\%$.

18) Let the number be $x$. $4.5\%=0.045$, so $0.045x=2.7$. Then $x=2.7 \div 0.045=60$.

19) Let the starting amount be $x$. $35\%=0.35$, so $0.35x=28$. Then $x=28 \div 0.35=80$. Answer: $\$80$.

20) Let the number be $x$. $130\%=1.30$, so $1.30x=156$. Then $x=156 \div 1.30=120$.

How to solve percent problems