Percents above 100 and below 1
A puppy can end its first year more than twenty times as heavy as it was at birth. If a percent counts hundredths of a whole, how can one amount be more than a hundred percent of another, and what does a percent smaller than one mean?
Where you are 8 in Percent
Why this exists
A percent first appears as a part of a whole: 18 of 24 tulip bulbs, 7 of 20 players. A part can never be bigger than its whole, so those percents run from 0\% to 100\%.
Many comparisons are not a part against its whole, though. A town is compared with the same town twenty years ago, a puppy with itself at birth, one dog with another, an office tower with the shed beside it. The second amount is not a piece of the first, and it can easily be the larger. Saying that a school now has 162.5\% of the students it opened with puts the two sizes side by side in one number, just as 75\% did for a part.
At the other end, some parts are so small that a whole percent is too coarse to show them. In a town of 36{,}000 people, 1\% is already 360 people. A group of 27 is far smaller than that, and rounding it to 0\% would say it is not there at all. Percents below one, such as 0.5\% or 0.075\%, keep a group like that in view.
Neither needs a new idea. A percent counts hundredths, and nothing about hundredths stops at one hundred of them or insists on at least one.
Definition
Percent means “per hundred”, so a percent is a number of hundredths. For any number p \ge 0, whole or not, large or small,
The percent sign divides whatever stands in front of it by 100. When p is above 100 the result is above 1, as in 250\% = 2.5. When p is below 1 the result is below \displaystyle \frac{1}{100}, as in 0.5\% = 0.005.
Taking a percent of an amount A means multiplying by that number, as in Finding a percent of a quantity:
Saying that an amount B is p\% of an amount A, where A > 0, means \displaystyle B = \frac{p}{100} \times A. Solving for p gives the way Finding what percent one number is of another finds it:
The amount after the word “of” is the whole, the 100\%. When B is a part of A, p can be at most 100. When B is a separate amount, such as a later size, another animal or another building, p can be any number of zero or more.
Intuition
Picture 100\% as one ten-by-ten grid with all hundred of its squares shaded. Then 200\% is two whole grids, and 250\% is two whole grids and half of a third.
Counted as hundredths, 250 of them make 2.5 wholes. That is why 250\% of an amount is 2.5 times the amount: two full copies of it and half of another.
The grid works just as well in the other direction. One small square is 1\% of the grid. Half of one small square is 0.5\%, and a tenth of a small square would be 0.1\%. These are still percents, hundredths and parts of a hundredth of the whole.
So a percent is a multiplier written in hundredths. Above 100\% it makes an amount larger, exactly 100\% leaves it as it is, and below 100\% it makes it smaller. Below 1\% it leaves less than a hundredth of the amount.
Results
Every percent is a number. For every p \ge 0, \displaystyle p\% = \frac{p}{100}. Some values:
| Percent | Decimal | Fraction |
|---|---|---|
| 1{,}000\% | 10 | 10 |
| 250\% | 2.5 | \displaystyle \frac{5}{2} |
| 100\% | 1 | 1 |
| 1\% | 0.01 | \displaystyle \frac{1}{100} |
| 0.5\% | 0.005 | \displaystyle \frac{1}{200} |
| 0.1\% | 0.001 | \displaystyle \frac{1}{1000} |
Above, at and below the whole. For any amount A > 0:
- p\% of A is more than A exactly when p > 100;
- p\% of A equals A exactly when p = 100;
- p\% of A is less than A exactly when p < 100;
- p\% of A is less than one hundredth of A exactly when p < 1.
Comparing two amounts. For positive amounts A and B, B is more than 100\% of A exactly when B > A, and B is less than 1\% of A exactly when \displaystyle B < \frac{A}{100}.
Why it is true
The percent sign is shorthand for “hundredths”, and a count of hundredths is a number whatever the count is. Two hundred and fifty hundredths is \displaystyle \frac{250}{100} = 2.5, in the same way that twenty-five hundredths is 0.25. Nothing in the definition asks for the count to be at most 100 or at least 1.
To compare p\% of A with A itself, subtract and take out the common factor A, which is the distributive property:
Because A and 100 are both positive, the right side has the same sign as p - 100. It is positive when p > 100, zero when p = 100 and negative when p < 100, and a positive difference means p\% of A is the larger. The same step with one hundredth of A gives
which is negative exactly when p < 1.
For two amounts, B is p\% of A with \displaystyle p = \frac{B}{A} \times 100. Multiplying both sides of B > A by the positive number \displaystyle \frac{100}{A} keeps the inequality the same way round, so B > A exactly when \displaystyle \frac{B}{A} \times 100 > 100, that is, when p > 100. In the same way, \displaystyle B < \frac{A}{100} exactly when p < 1.
The grid picture is the distributive property too. Since 250 = 100 + 100 + 50,
two whole copies of A and half of another.
Worked examples
Writing large and small percents as numbers
Write each percent as a decimal and as a fraction in lowest terms.
(a) 250\% (b) 1{,}200\% (c) 0.5\% (d) \displaystyle \frac{3}{4}\%
SolutionEvery part uses the same fact: \displaystyle p\% = \frac{p}{100}, so divide by 100.
(a) \displaystyle 250\% = \frac{250}{100} = 2.5. Dividing top and bottom by 50 gives \displaystyle \frac{5}{2}. The percent is above 100, so the number is above 1.
(b) \displaystyle 1{,}200\% = \frac{1200}{100} = 12. Twelve whole hundreds make the whole number 12.
(c) \displaystyle 0.5\% = \frac{0.5}{100}. Multiplying top and bottom by 10 clears the decimal: \displaystyle \frac{5}{1000} = \frac{1}{200}. As a decimal, dividing 0.5 by 100 moves the point two places to the left, which needs two zeros in front: 0.5\% = 0.005. The percent is below 1, so the number is below \displaystyle \frac{1}{100}.
(d) \displaystyle \frac{3}{4}\% means \displaystyle \frac{3}{4} divided by 100, which is \displaystyle \frac{3}{400}. Since \displaystyle \frac{3}{4} = 0.75, the same percent is 0.75\%, and 0.75 \div 100 = 0.0075.
Check (b) the other way: 12 = 12 \times 100\% = 1{,}200\%.
One dog compared with another
Pip weighs 12 kg. Bruno weighs 250\% of what Pip weighs. How heavy is Bruno?
SolutionThe words “of what Pip weighs” make Pip’s weight the whole, the 100\%. Since 250\% = 2.5,
Bruno weighs 30 kg. The answer is more than Pip’s 12 kg, as it must be when the percent is above 100.
The figure shows the same answer split up: 200\% of 12 is two copies of Pip, 24 kg, and 50\% of 12 is half a copy, 6 kg, so Bruno weighs 24 + 6 = 30 kg.
Check: \displaystyle \frac{30}{12} \times 100 = 250, so 30 kg is 250\% of 12 kg.
Bruno is 30 - 12 = 18 kg heavier than Pip, and 18 is 150\% of 12. Measuring a difference against the amount it is compared with in this way is the subject of Percent increase and decrease; this lesson compares the two sizes themselves.
A school compared with its first year
Elmfield School opened with 640 students. This year it has 1{,}040. This year’s number of students is what percent of the number it opened with?
Solution“Of the number it opened with” makes 640 the whole. Divide and simplify, taking out a common factor of 80:
and 1.625 = 162.5\%. This year the school has 162.5\% of its opening number of students.
The percent is above 100 because this year’s number is the larger one. Read as a multiplier, the school is now \displaystyle 1\frac{5}{8} times the size it started.
Check: 1.625 \times 640 = 640 + 400 = 1040, since 0.625 \times 640 = 400.
A puppy’s first year
Astrid’s puppy weighed 0.4 kg at birth and 9.2 kg on its first birthday.
(a) Its weight on its birthday is what percent of its birth weight?
(b) At six months old it weighed 1{,}500\% of its birth weight. How heavy was it then?
Solution(a) The birth weight, 0.4 kg, is the whole. Multiplying top and bottom by 10 clears the decimals:
and 23 = 2{,}300\%. On its birthday the puppy weighs 2{,}300\% of its birth weight. A percent this large is nothing new: it says the puppy is 23 times as heavy.
(b) \displaystyle 1{,}500\% = \frac{1500}{100} = 15, so the weight at six months was
The puppy weighed 6 kg at six months.
Check: 6 kg lies between 0.4 kg and 9.2 kg, just as 1{,}500\% lies between 100\% and 2{,}300\%.
A small group in a large town
The made-up town of Larkhill has 36{,}000 people.
(a) How many people is 0.5\% of the town?
(b) Of the town’s people, 27 are more than 100 years old. What percent of the town is that?
Solution(a) 0.5\% = 0.005, so
That is 180 people. Another way: 1\% of 36{,}000 is 360, and 0.5\% is half of 1\%, so it is half of 360.
(b) The whole is the town, 36{,}000. Since 27 is far less than 1\% of the town, 360 people, the percent will be below 1:
So 27 people are 0.075\% of the town.
Check: 0.075\% = 0.00075, and 0.00075 \times 36{,}000 = 27.
Rounded to a whole percent, 0.075\% would be 0\%, which would make 27 real people vanish. Percents below one keep their decimal places, here exactly 0.075\%, or about 0.08\%.
A tower and a shed
An office tower is 250 m tall. A garden shed beside it is 2 m tall.
(a) The shed’s height is what percent of the tower’s?
(b) The tower’s height is what percent of the shed’s?
Solution(a) The tower is the whole:
The shed is 0.8\% of the tower’s height, less than one hundredth of it.
(b) Now the shed is the whole:
The tower is 12{,}500\% of the shed’s height, 125 times as tall.
The two answers are tied together. Their product is 0.8 \times 12{,}500 = 10{,}000, and that is no accident: for any two positive amounts,
So whenever one amount is below 1\% of another, the other is above 10{,}000\% of the first. A percent far below one and a percent far above one hundred describe the same pair of sizes, seen from opposite ends.
Chaining comparisons between bridges
Three made-up bridges are compared by the length of their main span. The Harbour Bridge’s span is 160\% of the River Bridge’s, and the River Bridge’s span is 75\% of the Gorge Bridge’s. The Harbour Bridge’s span is what percent of the Gorge Bridge’s?
SolutionWrite each “of” as a multiplication. Let G be the Gorge Bridge’s span. Then the River Bridge’s span is 0.75G, and the Harbour Bridge’s span is 160\% of that:
Regrouping the product is the associative property. Since 1.2 = 120\%, the Harbour Bridge’s span is 120\% of the Gorge Bridge’s.
The two percents were measured against different wholes, 160\% against the River Bridge and 75\% against the Gorge Bridge, so they are combined by multiplying the multipliers, not by adding or subtracting the percents.
Check with lengths: if the Gorge Bridge’s span is 400 m, the River Bridge’s is 0.75 \times 400 = 300 m and the Harbour Bridge’s is 1.6 \times 300 = 480 m, and \displaystyle \frac{480}{400} = 1.2.
Boundaries
- A part cannot pass 100\% of its own whole. If a school has 500 students, at most 500 of them can be anywhere at once, which is 100\%. A percent above 100 needs two separate amounts, such as a later size and an earlier one, or one animal and another, where the amount after “of” is not something the other amount is a piece of.
- “Of” and “more than” are different comparisons. B is 250\% of A means B = 2.5A. B is 250\% more than A means B is A together with 250\% of A, which is 350\% of A. The second measures the difference, which is the subject of Percent increase and decrease.
- The whole cannot be zero. Finding \displaystyle p = \frac{B}{A} \times 100 divides by A, so A must be positive. No amount is any percent of nothing.
- Percents below one need their decimal places. Rounded to a whole percent, 0.075\% becomes 0\%. The percent sign is what does the dividing by 100: 0.5\% is 0.005, while the plain decimal 0.5 is 50\%.
- Sizes give percents of zero or more. Every amount compared here is a size, a count, a height or a weight, so every percent is 0 or more. Changes, which can go down as well as up, are measured in Percent change.
Where this goes next
Percents above 100 are the everyday language of growth. Percent increase and decrease measures the change itself against where it started: the puppy’s gain of 8.8 kg in its first year is 2{,}200\% of its birth weight. The multiplier for a percent change turns that round and writes the new amount as one percent of the old, often a percent above 100, so that a rise is a single multiplication. Chaining two comparisons, as with the bridges, becomes Two percent changes one after the other, where the multipliers are multiplied rather than the percents added.
Working backwards to the whole, when the given amount is a percent of it, is Finding the whole from a part and a percent, and Solving percent problems with an equation or a proportion writes every percent question as one equation, \displaystyle B = \frac{p}{100} \times A, in which p may be any size. Small percents come back in Percent error, where a careful measurement may be off by well under 1\% of the true value.
Exercises
Percents as numbers
- Write 300\% as a decimal.
- Write 145\% as a decimal and as a fraction in lowest terms.
- Write 0.2\% as a decimal and as a fraction in lowest terms.
- Write 0.04\% as a decimal and as a fraction in lowest terms.
- Write \displaystyle \frac{1}{4}\% as a decimal and as a fraction in lowest terms.
- Write each number as a percent: (a) 3.6 (b) 0.007 (c) \displaystyle \frac{9}{4} (d) \displaystyle \frac{1}{1000}
Taking a percent of an amount
- Find 175\% of 48.
- A lamb weighed 4.5 kg at birth. Four months later it weighs 600\% of its birth weight. How heavy is it at four months?
- A made-up elephant calf weighs 120 kg. An adult elephant in the same herd weighs 4{,}500\% of the calf’s weight. How heavy is the adult?
- The made-up town of Brackwater has 18{,}400 people. How many people is 0.5\% of the town?
- A made-up steel bridge is 1{,}500 m long. How many meters is 0.03\% of its length?
Finding the percent
- A village had 2{,}500 people in 1990 and has 3{,}100 now. Today’s population is what percent of the 1990 population?
- Leif’s tomato plant was 15 cm tall on 1 May and 87 cm tall on 1 July. Its July height is what percent of its May height?
- A horse weighs 540 kg and a pony weighs 180 kg. (a) The horse’s weight is what percent of the pony’s? (b) The pony’s weight is what percent of the horse’s?
- In a made-up city of 48{,}000 people, 30 people keep bees. What percent of the city is that?
- A made-up road bridge is 1{,}600 m long and a footbridge in a park is 8 m long. (a) The footbridge’s length is what percent of the road bridge’s? (b) The road bridge’s length is what percent of the footbridge’s?
Working back to the whole
- Ottilie’s sunflower is 180 cm tall, which is 900\% of its height on 1 June. How tall was it on 1 June?
- A town has 6{,}300 people, which is 140\% of its population in 1980. What was its population in 1980?
- Six cows are 0.4\% of a farm’s herd. How many cows are in the herd?
Comparing and combining
- Dog A weighs 125\% of what dog B weighs, and dog B weighs 80\% of what dog C weighs. Dog A’s weight is what percent of dog C’s?
- Put these in order from smallest to largest: 1.2, 0.8\%, 120\%, 12\%, 0.08.
- A town had 5{,}000 people in 1950. One report says the town now has 200\% of its 1950 population. Another says it has 200\% more than its 1950 population. How many people does each report describe?
Reasoning
- Without working out either answer, explain why 340\% of 50 must be more than 50, and why 0.6\% of 50 must be less than 0.5.
- Could 120\% of a school’s students be in its hall at one time? Could a sunflower be 120\% of its height a week earlier? Explain the difference.
- Show that if B is p\% of A, where A, B and p are all positive, then A is \displaystyle \frac{10{,}000}{p}\% of B. Use this to check the two answers in the tower and shed example.
Answers
- 3. Divide by 100: \displaystyle \frac{300}{100} = 3.
- \displaystyle 1.45 = \frac{29}{20}. \displaystyle \frac{145}{100}, and dividing top and bottom by 5 gives \displaystyle \frac{29}{20}.
- \displaystyle 0.002 = \frac{1}{500}. \displaystyle \frac{0.2}{100} = \frac{2}{1000} = \frac{1}{500}.
- \displaystyle 0.0004 = \frac{1}{2500}. \displaystyle \frac{0.04}{100} = \frac{4}{10{,}000} = \frac{1}{2500}.
- \displaystyle 0.0025 = \frac{1}{400}. \displaystyle \frac{1}{4} divided by 100 is \displaystyle \frac{1}{400}; also \displaystyle \frac{1}{4}\% = 0.25\% and 0.25 \div 100 = 0.0025.
- (a) 360\%, since 3.6 \times 100 = 360. (b) 0.7\%, since 0.007 \times 100 = 0.7. (c) 225\%, since \displaystyle \frac{9}{4} = 2.25. (d) 0.1\%, since \displaystyle \frac{1}{1000} = 0.001 and 0.001 \times 100 = 0.1.
- 84. 1.75 \times 48 = 48 + 36 = 84.
- 27 kg. 600\% = 6, and 6 \times 4.5 = 27.
- 5{,}400 kg. 4{,}500\% = 45, and 45 \times 120 = 5400.
- 92 people. 0.005 \times 18{,}400 = 92; or 1\% is 184 and half of that is 92.
- 0.45 m. 0.03\% = 0.0003, and 0.0003 \times 1500 = 0.45.
- 124\%. \displaystyle \frac{3100}{2500} = 1.24.
- 580\%. \displaystyle \frac{87}{15} = 5.8.
- (a) 300\%, since \displaystyle \frac{540}{180} = 3. (b) \displaystyle 33\frac{1}{3}\%, since \displaystyle \frac{180}{540} = \frac{1}{3} and \displaystyle \frac{1}{3} \times 100 = \frac{100}{3} = 33\frac{1}{3}.
- 0.0625\%. \displaystyle \frac{30}{48{,}000} \times 100 = \frac{3000}{48{,}000} = \frac{1}{16} = 0.0625.
- (a) 0.5\%, since \displaystyle \frac{8}{1600} = \frac{1}{200} = 0.005. (b) 20{,}000\%, since \displaystyle \frac{1600}{8} = 200. As in the tower and shed example, 0.5 \times 20{,}000 = 10{,}000.
- 20 cm. 900\% = 9, so 9 \times h = 180 and h = 20. Check: 9 \times 20 = 180.
- 4{,}500 people. 1.4 \times P = 6300, so P = 6300 \div 1.4 = 4500.
- 1{,}500 cows. 0.4\% = 0.004, so 0.004 \times n = 6 and n = 6 \div 0.004 = 1500.
- 100\%. A weighs 1.25 \times (0.8C) = (1.25 \times 0.8)C = 1 \times C, so dogs A and C weigh the same.
- 0.8\% < 0.08 < 12\% < 1.2 = 120\%. As decimals the five are 1.2, 0.008, 1.2, 0.12 and 0.08, and 1.2 and 120\% are equal.
- 10{,}000 and 15{,}000. 200\% of 5{,}000 is 2 \times 5000 = 10{,}000. “200\% more” adds 200\% of 5{,}000 to the 5{,}000: 5000 + 10{,}000 = 15{,}000.
- 340\% = 3.4, which is more than 1, and multiplying a positive amount by a number more than 1 makes it larger. 0.6\% = 0.006, which is less than 0.01, so 0.6\% of 50 is less than 1\% of 50, and 1\% of 50 is 0.5.
- No for the hall: the students in the hall are a part of the school’s students, so they are at most all of them, 100\%. Yes for the sunflower: its height now and its height a week ago are two separate amounts, and 120\% says only that it is now 1.2 times as tall.
- \displaystyle B = \frac{p}{100}A, and dividing both sides by \displaystyle \frac{p}{100} gives \displaystyle A = \frac{100}{p}B. Since \displaystyle \frac{100}{p} = \frac{10{,}000/p}{100}, A is \displaystyle \frac{10{,}000}{p}\% of B. With the shed at p = 0.8, \displaystyle \frac{10{,}000}{0.8} = 12{,}500, which matches the tower’s 12{,}500\%.