Estimating Percents with Benchmarks | Grade 7 Math | LevelKite

Sometimes you need a percent of a number fast, and a close answer is good enough. An estimate is a value near the exact answer that you find with easier numbers.

A percent is a rate per 100 of a stated positive whole. The whole is the amount you take the percent of. In $26\%$ of $80$ , the whole is $80$ , and $26\%$ means $\frac{26}{100}$ of that whole.

In this lesson you will estimate a percent of a number and explain how close your estimate is to the exact value.

A benchmark percent is a percent that matches a simple fraction of the whole, so you can find it in your head. You already know these conversions:

$50\% = \frac{50}{100} = \frac{1}{2} = 0.5$
$25\% = \frac{25}{100} = \frac{1}{4} = 0.25$
$10\% = \frac{10}{100} = \frac{1}{10} = 0.1$
$1\% = \frac{1}{100} = 0.01$

Each fraction or decimal is a multiplier: the number you multiply the stated positive whole by to get that percent of it. Multiplying by $\frac{1}{4}$ gives the same result as dividing by $4$ , and in the same way every benchmark becomes a quick division. The table shows all four benchmarks for one stated whole, $240$ .

Benchmark percents of the same whole, 240

Benchmark percent	Fraction multiplier	Decimal multiplier	Shortcut	Value when the whole is 240
$50\%$	$\frac{1}{2}$	$0.5$	Divide by $2$	$240 \div 2 = 120$
$25\%$	$\frac{1}{4}$	$0.25$	Divide by $4$ : halve, then halve again	$240 \div 4 = 60$
$10\%$	$\frac{1}{10}$	$0.1$	Divide by $10$	$240 \div 10 = 24$
$1\%$	$\frac{1}{100}$	$0.01$	Divide by $100$	$240 \div 100 = 2.4$

Every row of the table uses the same whole, so the rows fit together. The value $60$ is half of $120$ because the multiplier $\frac{1}{4}$ is half of $\frac{1}{2}$ . The value $2.4$ is one tenth of $24$ because $\frac{1}{100}$ is one tenth of $\frac{1}{10}$ .

You can also add benchmarks of the same stated positive whole to reach other percents, because the multipliers add: $\frac{50}{100} + \frac{10}{100} = \frac{60}{100}$ . Percents of different wholes cannot be added this way. For the whole $240$ , the sum looks like this:

60\% \text{ of } 240 = 50\% \text{ of } 240 + 10\% \text{ of } 240 = 120 + 24 = 144

Key concept

Estimate with benchmark multipliers

Let $W$ be the whole: a stated positive number, with its unit if it has one. Let $p$ be the positive number in front of the percent sign. Because a percent is a rate per 100 of that whole, $p\%$ of $W$ equals $\frac{p}{100}$ times $W$ .

To estimate $p\%$ of $W$ :

Replace $p\%$ with a nearby benchmark ( $1\%$ , $10\%$ , $25\%$ or $50\%$ ) or a sum of benchmarks.
If it helps, replace $W$ with a nearby positive number that is easy to divide. Take every benchmark of that one whole.
Multiply, then say which way the rounding moved the answer and by about how much.

p\% \text{ of } W = \frac{p}{100} \times W \quad \text{for a stated positive whole } W

Worked example

Round the percent only

Estimate $26\%$ of $80$ .

Name the whole and pick a benchmark. The whole is 80, a stated positive number. The percent 26% is close to the benchmark 25%, whose multiplier is one fourth, so estimate with 25% of the same whole.
$26\% \text{ of } 80 \approx 25\% \text{ of } 80$
Multiply the whole by one fourth. That is the same as dividing by 4: half of 80 is 40, and half of 40 is 20.
$\frac{1}{4} \times 80 = 80 \div 4 = 20$
Explain the closeness. The exact amount, 26% of 80, is 25% of 80 plus 1% of 80, because the multipliers add for the same whole. The estimate leaves out that 1% of 80, so it is below the exact value by that amount. Find it with the decimal multiplier 0.01.
$0.01 \times 80 = 0.8$
Add the gap to the estimate to get the exact value, so you can see how close the estimate is.
$20 + 0.8 = 20.8$

Answer

$26\%$ of $80$ is about $20$ . The estimate is $0.8$ below the exact value, $20.8$ , because $25\%$ of $80$ leaves out $1\%$ of $80$ .

Which way did your rounding push the answer? These rules hold for a positive percent of a stated positive whole. The reason is that the answer is a positive multiplier times a positive whole, so making either factor larger makes the product larger.

Change one number only, either the percent or the whole. Rounding it up gives an overestimate. Rounding it down gives an underestimate.
Round both the same way. Both changes push the estimate in the same direction, so their sizes add.
Round them opposite ways. The changes work against each other, so find the size of each to see which is larger.

Benchmarks give you the size of a change, just as the first example measured its gap as $1\%$ of the whole.

Worked example

Round the whole only

A bag holds $795$ grams (g) of trail mix, and $35\%$ of that mass is raisins. About how many grams of raisins are in the bag?

Name the whole and round it. The whole is the mass of the mix, 795 grams, a stated positive amount. Round it up to 800 grams, which is easy to divide by 4 and by 10. From here on, take every benchmark of this one rounded whole.
$35\% \text{ of } 795 \text{ g} \approx 35\% \text{ of } 800 \text{ g}$
Split the percent into benchmarks: 35% is 25% plus 10%. Find each part of 800 grams with its multiplier, one fourth and one tenth.
$\frac{1}{4} \times 800 \text{ g} = 200 \text{ g}, \qquad \frac{1}{10} \times 800 \text{ g} = 80 \text{ g}$
Add the two parts. Adding is valid because both are percents of the same 800 grams.
$200 \text{ g} + 80 \text{ g} = 280 \text{ g}$
Explain the closeness. Only the whole changed, and you rounded it up. Since 800 = 795 + 5, 35% of 800 is 35% of 795 plus 35% of 5. So the estimate is above the exact value by 35% of 5 grams. Use the same two benchmarks on 5 grams.
$\frac{1}{4} \times 5 \text{ g} + \frac{1}{10} \times 5 \text{ g} = 1.25 \text{ g} + 0.5 \text{ g} = 1.75 \text{ g}$

Answer

About $280$ grams of the mix is raisins. This is an overestimate by $1.75$ grams, so the exact amount is $280 - 1.75 = 278.25$ grams.

A common slip is to reason that because you divide by $10$ to find $10\%$ , you must divide by $25$ to find $25\%$ . For the whole $800$ grams from the last example, that gives $800 \div 25 = 32$ grams.

It fails because dividing by $25$ multiplies the whole by $\frac{1}{25}$ , and $\frac{1}{25} = \frac{4}{100}$ . So $32$ grams is only $4\%$ of $800$ grams. The divisor comes from the simplified fraction, not from the percent number: $10\% = \frac{1}{10}$ , so you divide by $10$ , but $25\% = \frac{1}{4}$ , so you divide by $4$ and get $200$ grams.

A check for any stated positive whole: a larger percent of the same whole is a larger amount. So $25\%$ of $800$ must be more than $10\%$ of $800$ , which is $80$ . The value $32$ is not.

Worked example

Round both numbers

Estimate $48\%$ of $612$ .

Round both numbers. The whole is 612, a stated positive number. Round the percent up to the benchmark 50% and the whole down to 600, which is easy to halve. Take the benchmark of that rounded whole.
$48\% \text{ of } 612 \approx 50\% \text{ of } 600$
Multiply the rounded whole by the multiplier one half.
$\frac{1}{2} \times 600 = 300$
Measure the percent change on the stated whole. Going from 48% to 50% of 612 adds 2% of 612. Find 1% by dividing by 100, then double it.
$612 \div 100 = 6.12, \qquad 2 \times 6.12 = 12.24$
Measure the whole change. Since 612 = 600 + 12, 50% of 612 is 50% of 600 plus 50% of 12. Dropping the 12 removes half of 12.
$\frac{1}{2} \times 12 = 6$
Compare the changes. They work against each other, and the increase is larger than the decrease, so the estimate is above the exact value by their difference.
$12.24 - 6 = 6.24$

Answer

$48\%$ of $612$ is about $300$ . This is an overestimate by $6.24$ , so the exact value is $300 - 6.24 = 293.76$ .

Takeaway. To estimate a percent of a stated positive whole, lean on the benchmarks: $50\%$ is one half, $25\%$ is one fourth, $10\%$ is one tenth and $1\%$ is one hundredth of that same whole. Round the percent, the whole, or both so the numbers fit those multipliers, then multiply. Finish by saying which way the rounding pushed your answer and by about how much. That closeness statement is what makes an estimate useful.