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Finding the whole from a part and a percent

You can see how much you have, and you know what percent of the total it is. How big is the total?

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When you reach for it

A percent situation has three numbers in it: a whole, a part of that whole, and the percent the part is of the whole. Know any two and the third follows, and the three percent questions are the three ways of choosing which one is missing. This lesson is the one that runs backwards. You know the part and its percent, and the whole is what you want.

It turns up whenever a gauge or a progress bar shows a percent and you can see only how much has arrived so far. A download bar reads 20\% after 46 megabytes. A charity’s thermometer chart stands at 65\% with 1{,}560 dollars raised. A phone battery reaches 15\% after 18 minutes on the charger. Each time the question is the same: how big is the thing the percent is a percent of? How big is the file, what is the target, how long does a full charge take?

The other two questions run forwards. Finding a percent of a quantity starts from the whole and multiplies. Finding what percent one number is of another starts from the part and the whole and divides. Here the whole is the unknown, so the move is to undo: work back from the slice you have to the 100\% you want.

This lesson takes two days. Part 1 reasons on a double number line, scaling the part up to 100\% the way you scale any ratio. Part 2 writes the same fact as an equation, part equals percent times whole, and solves it, which works for any numbers at all.

The procedure

Everything rests on what a percent says. A Percent is a number of hundredths, so “12 is 40\% of the whole” means that 12 is 40 hundredths of the whole:

12 = \frac{40}{100} \times \text{whole}

Put another way, the part and the whole are in the same ratio as 40 and 100. The whole is always 100\% of itself, so finding the whole means finding the amount that goes with 100\%.

Part 1: scaling up to 100%

Line up two scales, amounts on one and percents on the other, so that 0 sits against 0\% and the whole sits against 100\%. Every amount on the top line matches the percent below it, and all the matching pairs are in the same ratio. So you may multiply or divide both numbers of a pair by the same non-zero number and land on another matching pair, because that is exactly what equivalent ratios are. The aim is to reach the pair with 100\% in it.

For “12 is 40\% of what?”, keep the pairs in a ratio table:

amount12330
percent40\%10\%100\%

Dividing both numbers by 4 gives 3 against 10\%. Multiplying both by 10 gives 30 against 100\%. The whole is 30. Checking forwards, 40\% of 30 is 0.4 \times 30 = 12, as it should be.

How to choose the steps depends on the percent. There are three good ways, and they always agree.

A benchmark percent. Some percents are simple fractions: \displaystyle 50\% = \frac{1}{2}, \displaystyle 25\% = \frac{1}{4}, \displaystyle 20\% = \frac{1}{5}, \displaystyle 10\% = \frac{1}{10} (see Benchmark percents and mental percent). If the part is a quarter of the whole, the whole is four quarters, so it is four times the part. In general, if the percent goes into 100\% some number of times, multiply the part by that number.

A friendly chunk. When the percent is not a benchmark, divide the pair down to a percent that goes evenly into 100, such as 5\%, 10\% or 20\%, then multiply up. That is the table above, with 10\% as the chunk.

Through 1\%. Dividing the pair by the percent itself always lands on 1\%. For 12 and 40\%, 1\% goes with 12 \div 40 = 0.3. Then multiplying by 100 reaches 100\%: 0.3 \times 100 = 30. This is the unit rate idea, the amount for each single percent. It works for every percent, at the price of decimals in the middle.

All three do the same thing in the end: divide the part by the percent and multiply by 100. With the percent written as p\%,

\text{whole} = \frac{\text{part}}{p} \times 100

Is the answer sensible? When the percent is below 100\%, the part is only a slice, so the whole must come out bigger than the part. Here, 40\% is a little under half, so 12 is a little under half the whole, and the whole is a little over 24. The answer 30 fits.

Part 2: an equation for the whole

The double number line is quickest with friendly numbers. For any numbers at all, write the statement as an equation. Call the unknown whole w and write the percent as a decimal, since \displaystyle p\% = \frac{p}{100}:

\text{part} = \frac{p}{100} \times w

For “12 is 40\% of w” this is 12 = 0.40w. The unknown has been multiplied by 0.40, so divide both sides by 0.40. That is allowed because dividing both sides of an equation by the same non-zero number keeps it true (the division property of equality):

w = \frac{12}{0.40} = 30

To divide by the decimal, multiply both numbers by 100 first: 12 \div 0.40 = 1200 \div 40 = 30. That is the formula from Part 1 in a different order: divide by \displaystyle \frac{40}{100} and you have divided by 40 and multiplied by 100.

The same fact as a proportion. “12 is 40\% of w” also says that the part is to the whole as 40 is to 100:

\frac{12}{w} = \frac{40}{100}

Solving the proportion with cross products, 40w = 12 \times 100 = 1200, so w = 30 again. The equation and the proportion are one statement written two ways, and Solving percent problems with an equation or a proportion uses that one statement for all three percent questions.

Pair each amount with its own percent. The part in the equation must be the amount the percent describes. A download bar that reads 64\% with 108 megabytes still to come gives you the rest of the file, not the part that has arrived. The rest is 100\% - 64\% = 36\% of the file, so 108 = 0.36w and the file is w = 108 \div 0.36 = 300 megabytes. Before you divide, ask which slice the amount you know is, and what percent that slice is.

Worked examples

Examples 1 to 3 go with Part 1, the double number line. Examples 4 to 7 go with Part 2, the equation.

Example 1

Three benchmark percents

(a) Linnea is saving for a tent. She has saved 45 dollars, and her savings tracker says that is 25\% of her goal. What is the goal?

(b) A farm’s water tank is 75\% full and holds 2{,}400 liters. How much does it hold when full?

(c) A download bar reads 20\% after 46 megabytes (MB) of a file have arrived. How big is the file?

Solution

(a) 25\% is a quarter, so 45 dollars is a quarter of the goal. The whole goal is four quarters:

4 \times 45 = 180

The goal is 180 dollars. In the ratio table, both numbers of the pair 45 and 25\% have been multiplied by 4, giving 180 and 100\%.

(b) 75\% is three quarters. Divide the pair by 3 to get one quarter, then multiply by 4:

liters2{,}4008003{,}200
percent75\%25\%100\%

The full tank holds 3{,}200 liters. That is more than 2{,}400, as it must be, since the tank was not yet full.

(c) 20\% is a fifth, and there are five fifths in the whole: 5 \times 46 = 230. The file is 230 MB.

Example 2

A battery on the charger

A phone with a flat battery is put on to charge. After 18 minutes the screen shows 15\%. If it goes on charging at the same rate, how long does a full charge take?

Solution

Here the amount is a time rather than a quantity of something, and that is fine. Charging at the same rate means every minute adds the same charge, so minutes and percent stay in one ratio and the double number line applies: 18 minutes against 15\%, and the full charge against 100\%.

15 does not go into 100 evenly, but 5 does, and 15 = 3 \times 5. Divide the pair by 3: 6 minutes against 5\%. There are 20 fives in 100, so multiply by 20:

6 \times 20 = 120
0
60
\mathbf{18}
\mathbf{120}
0\%
50\%
\mathbf{15\%}
\mathbf{100\%}
 minutes charge
Figure 1The phone in Example 2. Every 5\% of charge takes 6 minutes, so 15\% sits against 18 minutes and 100\% against 120.

A full charge takes 120 minutes, which is 2 hours. As a check, 15\% is a little more than a seventh of 100\%, so the full time should be a little under 7 \times 18 = 126 minutes, and 120 is.

Example 3

A fundraising thermometer

A community garden is raising money for a greenhouse, and it shows its progress on a chart drawn as a thermometer. The fill has reached 65\% of the way up, and the label beside it says 1{,}560 dollars.

0\%
50\%
100\%
\mathbf{65\%}
1{,}560\ \text{dollars}
\text{target}
Figure 2The chart in Example 3. The fill is 65\% of the way to the top, and the top is the target, the whole.

(a) What is the target? (b) How much more is still to be raised?

Solution

(a) 65\% is not a benchmark, so go through 1\%. Dividing the pair by 65:

1{,}560 \div 65 = 24

so each 1\% of the target is 24 dollars, and 100\% is 24 \times 100 = 2{,}400 dollars. A friendly chunk works too. Both 65 and 1{,}560 divide by 13, giving 120 dollars against 5\%, and 20 of those make 2{,}400:

dollars1{,}560241202{,}400
percent65\%1\%5\%100\%

The target is 2{,}400 dollars. Checking forwards, 0.65 \times 2{,}400 = 1{,}560.

(b) Still to raise: 2{,}400 - 1{,}560 = 840 dollars. That is the empty part of the tube, the other 35\%, and indeed 35 \times 24 = 840.

Example 4

A reservoir, by equation and by proportion

A reservoir holds 54 million cubic meters of water, which is 72\% of its capacity. What is its capacity?

Solution

Let w be the capacity in millions of cubic meters. The water held is 72\% of it:

54 = 0.72w

Divide both sides by 0.72, which undoes the multiplication:

w = \frac{54}{0.72} = \frac{5400}{72} = 75

The middle step multiplies the top and the bottom by 100 so that the divisor is a whole number; the value of the fraction does not change.

The proportion says the same thing. The water is to the capacity as 72 is to 100:

\frac{54}{w} = \frac{72}{100}

Cross products give 72w = 5400, so w = 75.

The double number line would have found it as well, with a less obvious chunk: 54 and 72 are both multiples of 18, so dividing the pair by 18 gives 3 against 4\%, and 25 of those make 75 against 100\%.

The capacity is 75 million cubic meters. Check: 0.72 \times 75 = 54.

Example 5

A percent with a decimal in it

A file is 37.5\% downloaded, and 159 MB have arrived. How big is the file?

Solution

37.5\% is 0.375 as a decimal, so 159 = 0.375w and

w = \frac{159}{0.375} = \frac{159{,}000}{375} = 424

There is a neater route if you know that \displaystyle 37.5\% = \frac{375}{1000} = \frac{3}{8}. Then 159 MB is three eighths of the file, one eighth is 159 \div 3 = 53, and the whole file is 8 \times 53 = 424 MB. Both routes agree, as they must, because 0.375 and \displaystyle \frac{3}{8} are the same number.

The file is 424 MB. Check: 0.375 \times 424 = 159.

Example 6

The amount you know is the rest

Bastien is saving for a telescope. His savings tracker shows 62\% of the goal saved, and he still needs 114 dollars. What is the goal, and how much has he saved?

Solution

The 114 dollars is not the saved part. It is what is left, so it goes with the percent that is left:

100\% - 62\% = 38\%
\text{goal} = 100\%
62\%\ \text{saved}
38\%\ \text{to go}
\mathbf{114}\ \text{dollars}
Figure 3The goal in Example 6. The 114 dollars still needed is the unshaded piece, so it goes with 38\%, not with 62\%.

Now the amount and its percent match: 114 is 38\% of the goal w.

\begin{aligned} 114 &= 0.38w \\ w &= \frac{114}{0.38} = \frac{11{,}400}{38} = 300 \end{aligned}

The goal is 300 dollars, and he has saved 300 - 114 = 186 dollars.

Check with the other percent: 62\% of 300 is 0.62 \times 300 = 186, the same amount, and the two pieces make the whole: 186 + 114 = 300.

Example 7

Past the target

A fundraiser for a playground goes past its target. Its thermometer chart reads 120\%, with 5{,}040 dollars raised. What was the target?

Solution

Nothing about the method needs the percent to be under 100. The money raised is 120\% of the target w, and 120\% = 1.2:

\begin{aligned} 5{,}040 &= 1.2w \\ w &= \frac{5{,}040}{1.2} = \frac{50{,}400}{12} = 4{,}200 \end{aligned}

The target was 4{,}200 dollars. This time the whole is smaller than the part, because 120\% is the whole and a fifth more. As a check, a fifth of 4{,}200 is 840, and 4{,}200 + 840 = 5{,}040. Percents above 100 have their own lesson.

Boundaries

The percent must not be 0\%. 0\% of every whole is 0. So “8 is 0\% of what?” has no answer, and “0 is 0\% of what?” is true of every whole and picks out none. In the equation this is the rule that you cannot divide by zero.

The amount and the percent must describe the same slice. If you know what is left, pair it with the percent left, as in Example 6. If you know what has arrived, pair it with the percent shown.

The whole is the amount that the percent is “of”. It is the 100\% amount, whatever the story calls it: the target, the capacity, the full charge, the file.

Scaling needs a steady rate. On a double number line every pair must be in the same ratio. When the amount is a time, as with the battery in Example 2, that means the charge rises by the same amount every minute. If it does not, the answer is only an estimate.

A rounded percent gives an approximate whole. If a gauge shows 33\% when the true share is \displaystyle \frac{1}{3}, a whole found from 33\% is close but not exact.

The size of the percent decides which way the answer goes. Below 100\% the whole is bigger than the part, at exactly 100\% they are equal, and above 100\% the whole is smaller, as in Example 7. A percent below 1\% is no different: the whole is then more than a hundred times the part.

Where this goes next

One equation, part \displaystyle = \frac{p}{100} \times whole, answers all three percent questions; this lesson is the case where the whole is the letter to solve for. The same backwards move works with a fraction in place of a percent, in finding the whole from a fraction: “12 is \displaystyle \frac{2}{5} of what?” is the same question as “12 is 40\% of what?”.

It also opens the way to percent change by a multiplier. There, a new amount is a percent of the old one, often a percent above 100, and finding the old amount from the new one is exactly the division in this lesson.

Exercises

Part 1: benchmark percents

  1. 14 is 50\% of what number?
  2. 9 is 10\% of what number?
  3. 33 is 75\% of what number?
  4. 26 is 20\% of what number?

Part 1: on the double number line

  1. Kalani has saved 84 dollars, which is 40\% of her savings goal. What is the goal?
  2. A tablet with a flat battery is put on to charge. After 21 minutes it shows 35\%. At the same rate, how long does a full charge take?
  3. A water tower is 15\% full and holds 27{,}000 liters. What is its capacity?
  4. Six minutes after a download starts, its bar reads 8\%. At the same rate, how long does the whole download take?
  5. A camera battery charges from empty at a steady rate and reaches 7\% after 10 minutes. How long does a full charge take, to the nearest minute?

Part 2: by equation or proportion

  1. 54 is 36\% of what number?
  2. 7.2 is 4.5\% of what number?
  3. 2.4 is 0.8\% of what number?
  4. 91 is 130\% of what number?
  5. A reservoir at 84\% of its capacity holds 6.3 million cubic meters of water. What is its capacity?
  6. A school’s fundraising thermometer shows 3{,}915 dollars raised, which is 87\% of the target. What is the target?

Part 2: which amount goes with which percent

  1. A download is 45\% complete, and 286 MB are still to come. How big is the file?
  2. A rain-water tank is 70\% full, and 69 more liters would fill it. What is its capacity?
  3. Soraya has saved 391 dollars, which is 115\% of her savings goal. What was the goal, and by how much has she passed it?
  4. A phone’s battery shows 18\%, and the phone estimates 2.7 hours of use left. If it uses charge at a steady rate, how many hours would a full battery last?
  5. Tank A is 40\% full and holds 900 liters. Tank B is 75\% full and holds 1{,}500 liters. Which tank holds more when full, and by how much?

Reasoning

  1. Without finding w, explain how you can tell that “39 is 60\% of w” gives a w bigger than 39, while “39 is 150\% of w” gives a w smaller than 39.
  2. Use a double number line to explain why dividing the part by p and then multiplying by 100 gives the whole.
  3. Explain why the question “8 is 0\% of what number?” has no answer.
  4. Explain why dividing a part by 0.36 gives the same result as multiplying it by \displaystyle \frac{100}{36}, and why that matches the two routes of Part 1 and Part 2.
  5. Two downloads both show 25\%. One has 60 MB downloaded and the other 90 MB. Without finding either file’s size, explain why the second file is 1.5 times as big as the first.
Answers
  1. 28. 50\% is a half, so the whole is twice the part: 2 \times 14 = 28.
  2. 90. 10\% is a tenth, so the whole is ten times the part: 10 \times 9 = 90.
  3. 44. 75\% is three quarters. Divide by 3: 11 is 25\%. Multiply by 4: 44 is 100\%.
  4. 130. 20\% is a fifth, so the whole is 5 \times 26 = 130.
  5. 210 dollars. Divide the pair by 4: 21 dollars is 10\%. Multiply by 10: 210 dollars is 100\%.
  6. 60 minutes. Divide the pair by 7: 3 minutes is 5\%. Multiply by 20: 60 minutes is 100\%.
  7. 180{,}000 liters. Divide by 15: 1\% is 1{,}800 liters. Multiply by 100.
  8. 75 minutes. Divide by 8: 1\% takes 0.75 minutes. Multiply by 100.
  9. About 143 minutes. 1\% takes 10 \div 7 minutes, so 100\% takes \displaystyle \frac{1000}{7} = 142\frac{6}{7} minutes, which rounds to 143.
  10. 150. 54 = 0.36w, so w = 54 \div 0.36 = 5400 \div 36 = 150.
  11. 160. 7.2 = 0.045w, so w = 7.2 \div 0.045 = 7200 \div 45 = 160.
  12. 300. 0.8\% = 0.008, so w = 2.4 \div 0.008 = 2400 \div 8 = 300.
  13. 70. 91 = 1.3w, so w = 91 \div 1.3 = 910 \div 13 = 70. The whole is smaller than the part, since the percent is over 100.
  14. 7.5 million cubic meters. 6.3 = 0.84w, so w = 6.3 \div 0.84 = 630 \div 84 = 7.5.
  15. 4{,}500 dollars. 3{,}915 = 0.87w, so w = 391{,}500 \div 87 = 4{,}500.
  16. 520 MB. The 286 MB still to come is 100\% - 45\% = 55\% of the file, so w = 286 \div 0.55 = 520.
  17. 230 liters. The 69 liters fill the empty 30\%, so w = 69 \div 0.3 = 230.
  18. The goal was 340 dollars, and she has passed it by 51 dollars. 391 = 1.15w, so w = 391 \div 1.15 = 340, and 391 - 340 = 51.
  19. 15 hours. 2.7 hours goes with 18\%, so w = 2.7 \div 0.18 = 15.
  20. Tank A, by 250 liters. Tank A: 900 \div 0.4 = 2{,}250 liters. Tank B: 1{,}500 \div 0.75 = 2{,}000 liters. Tank B holds more water right now, but only because it is fuller: a part says nothing about the size of the whole until you know its percent.
  21. The whole is always 100\%. Since 60\% is less than 100\%, the 39 is only part of w, so w is more than 39. Since 150\% is more than 100\%, the 39 is more than all of w, so w is less than 39.
  22. On a double number line the part sits against p\% and the whole against 100\%, and all matching pairs are in one ratio. Dividing both numbers of the pair by p gives the amount against 1\%, which is \displaystyle \frac{\text{part}}{p}. Multiplying both by 100 then gives the amount against 100\%, which is the whole.
  23. 0\% of any whole is 0 \times whole = 0, never 8, so no whole works. In the equation 8 = 0 \times w you would have to divide by 0, which is not possible.
  24. \displaystyle 0.36 = \frac{36}{100}, and dividing by a fraction is the same as multiplying by its reciprocal, \displaystyle \frac{100}{36}. Multiplying by \displaystyle \frac{100}{36} is dividing by 36 and multiplying by 100, which is the Part 1 route through 1\%. Dividing by 0.36 is the Part 2 route of solving \text{part} = 0.36w. So both give the same whole.
  25. A part that is 25\% is a quarter of its file, so each file is 4 times its part. The second part is 1.5 times the first (90 = 1.5 \times 60), so the second file, 4 \times 90, is 1.5 times the first, 4 \times 60.
kg1128 · updated 2026-10-07 · CC-BY-SA-4.0