What a proportion says
Two jugs of lemonade were mixed from different amounts. How can you tell, without tasting, whether they will taste the same?
Where you are 3 in Proportion
Why this exists
Dalia mixes a jug of lemonade from 2 cups of concentrate and 3 cups of water. Tomas mixes his from 4 cups of concentrate and 6 cups of water. Tomas has more lemonade, but a glass from either jug tastes the same. What a glass tastes like does not depend on how much was made. It depends on how the concentrate compares with the water, and in both jugs that comparison is “2 cups of concentrate for every 3 cups of water”.
A ratio records that comparison, and one comparison has many names: 2 : 3, 4 : 6, 10 : 15, 1 : 1.5. Equivalent ratios showed how to make new names by scaling both parts. A proportion is the sentence that says two names belong to the same comparison:
Written as an equation, the claim can be tested, and it can be false. A painter matching a shade, a builder mixing concrete and a gardener diluting plant food all ask the same question: do these two sets of amounts make the same mixture? A proportion is how that question is written down, and when one of the four numbers is missing it becomes an equation to solve.
Definition
A proportion is a statement that two ratios are equivalent, that is, that they describe the same comparison. It is written either way:
Both are read “a is to b as c is to d”. In the fraction form b and d must not be 0.
Two ratios of positive numbers are equivalent when the second can be made from the first by multiplying both of its parts by the same positive number k:
This is the meaning from Equivalent ratios and ratio tables. The number k need not be whole: 4 : 6 becomes 6 : 9 when both parts are multiplied by k = \tfrac{3}{2}, and dividing both parts by 2 is multiplying them by k = \tfrac{1}{2}.
When the two ratios are equivalent, the statement is true and the ratios are called proportional. The statement 2 : 3 = 3 : 4 has the shape of a proportion but is false: no single number multiplies 2 to 3 and also 3 to 4. Those two ratios are not proportional, and we write 2 : 3 \ne 3 : 4. Deciding which case you are in is called testing whether two ratios are proportional.
A proportion compares like with like. The first part of each ratio must measure the same thing (cups of concentrate on both sides, say), the second parts must match too, and each matching pair must be in the same unit. 2 : 3 = 4 : 6 about concentrate and water is true; 2 : 3 = 6 : 4, with the second ratio listed water first, is a different and false statement.
Intuition
Pour two jugs of Dalia’s lemonade into one big bowl. Each jug tastes the same, so the bowl does too, and the bowl holds 4 cups of concentrate and 6 cups of water. That is Tomas’s jug.
Three copies give 6 : 9, ten copies give 20 : 30, and half a copy, if you could pour exactly half, gives 1 : 1.5. Every one of them is jug A made bigger or smaller, which is why multiplying both parts by the same number keeps the taste.
Adding is different. Put 1 more cup of concentrate and 1 more cup of water into jug A and you get 3 : 4, which cannot be split into copies of jug A.
A second way to see the difference is to ask how much water goes with one cup of concentrate, the unit rate. Jug A has \tfrac{3}{2} = 1.5 cups of water for each cup of concentrate. Tomas’s jug has \tfrac{6}{4} = 1.5 as well: same amount per cup, same taste. Jug C has \tfrac{4}{3}, about 1.33 cups, so each cup of concentrate is spread through less water and jug C tastes stronger.
Results
Four tests that always agree. For positive numbers a, b, c and d, each of these says exactly the same thing as “a : b = c : d is true”:
- A scale factor. One number k gives c = ka and d = kb. Equivalently, a : c = b : d: the first parts compare in the same way as the second parts.
- Lowest terms. The fractions \displaystyle \frac{a}{b} and \displaystyle \frac{c}{d} simplify to the same fraction.
- Per one. The amounts of the second part for one of the first part agree: \displaystyle \frac{b}{a} = \frac{d}{c}.
- Cross products. The products across the equals sign are equal:
Use whichever is quickest. A scale factor is easy to see when it is a whole number; the cross products work for any numbers, decimals and fractions included.
A table of mixtures is proportional when every row is equivalent to every other row. Equivalently, the quotient (second amount) \div (first amount) is the same number in every row. That shared number is the constant of proportionality.
More than two parts. a : b : c = d : e : f when one number k gives d = ka, e = kb and f = kc. Two of the three parts matching is not enough.
Adding the same amount to both parts of a : b gives an equivalent ratio only when a = b.
Why it is true
Start from a scale factor: c = ka and d = kb, with k positive.
The fraction \displaystyle \frac{c}{d} is \displaystyle \frac{ka}{kb}, and the common factor k cancels, so \displaystyle \frac{c}{d} = \frac{a}{b}. Equal fractions have the same lowest terms. In the same way \displaystyle \frac{d}{c} = \frac{kb}{ka} = \frac{b}{a}, so the per one amounts agree. And the cross products are
which are equal.
Now go back the other way. Suppose only that ad = bc. Take \displaystyle k = \frac{c}{a}, so that ka = c. Then
using bc = ad in the middle step. So k is a scale factor, and the ratios are equivalent. The cross products are therefore a complete test, not just a consequence.
The name comes from the fraction form. Multiply both sides of \displaystyle \frac{a}{b} = \frac{c}{d} by bd, which is not zero:
Each numerator ends up multiplied by the other side’s denominator, along the two diagonals of a cross.
For the last result, compare the cross products of a : b and (a + n) : (b + n) with n > 0:
These are equal exactly when an = bn, that is, when a = b.
Worked examples
Writing a proportion from a sentence
A shade of blue is made from 3 tins of blue paint for every 5 tins of white.
(a) Write the sentence “3 tins of blue to 5 tins of white make the same shade as 9 tins of blue to 15 tins of white” as a proportion, in both forms, and decide whether it is true.
(b) Rhea writes \displaystyle \frac{3}{5} = \frac{15}{9} for the same sentence. Is that the same statement?
Solution(a) Keep blue first and white second on both sides:
It is true. Multiplying both parts of 3 : 5 by k = 3 gives 9 : 15, so nine tins of blue with fifteen of white is three batches of the original shade.
(b) No. Her right side has white on top, so it compares white with blue while the left compares blue with white. As numbers, \displaystyle \frac{3}{5} is less than 1 and \displaystyle \frac{15}{9} is more than 1, so \displaystyle \frac{3}{5} \ne \frac{15}{9}: her statement is false. The sentence it was meant to record is true, which shows why the order of the parts has to match.
Testing pairs of ratios
Decide whether the two ratios are proportional.
(a) 4 : 10 and 6 : 15 \qquad (b) 8 : 12 and 10 : 16 \qquad (c) 2.5 : 4 and 10 : 16
Solution(a) No whole number takes 4 to 6, so use lowest terms:
The same fraction, so the ratios are proportional. The scale factor is k = \tfrac{6}{4} = 1.5, and indeed 10 \times 1.5 = 15.
(b) In lowest terms \displaystyle \frac{8}{12} = \frac{2}{3} and \displaystyle \frac{10}{16} = \frac{5}{8}. These are different fractions, so the ratios are not proportional. The cross products agree with that: 8 \times 16 = 128 but 12 \times 10 = 120.
(c) Multiplying both parts of 2.5 : 4 by k = 4 gives 10 : 16, so the ratios are proportional. Check with cross products: 2.5 \times 16 = 40 and 4 \times 10 = 40.
Which jugs taste the same
Three jugs of lemonade are mixed as follows.
| Jug | Concentrate (cups) | Water (cups) |
|---|---|---|
| P | 3 | 8 |
| Q | 6 | 15 |
| R | 4.5 | 12 |
Which jugs taste the same? Of any that differ, which is stronger?
SolutionFind the water for one cup of concentrate in each jug, the per one test:
P and R have the same amount of water per cup of concentrate, so 3 : 8 = 4.5 : 12 and they taste the same. Check with cross products: 3 \times 12 = 36 and 8 \times 4.5 = 36.
Q has only 2\tfrac{1}{2} cups of water for each cup of concentrate, less than the 2\tfrac{2}{3} in the others, so Q tastes stronger. Lowest terms tell the same story: \displaystyle \frac{3}{8} for P and R, \displaystyle \frac{2}{5} for Q.
Plant food, and matching units
Owen stirs 7.5 mL of liquid plant food into 2 L of water. Sanna stirs 12 mL into 3.2 L.
(a) Are the two mixtures equally strong?
(b) Sanna writes her mixture as 12 mL to 3200 mL and tests 7.5 : 2 = 12 : 3200. What goes wrong?
Solution(a) Keep plant food first and water second. The cross products of 7.5 : 2 and 12 : 3.2 are
They are equal, so the ratios are proportional and the mixtures are equally strong. Per liter, both come to 3.75 mL of plant food: \displaystyle \frac{7.5}{2} = 3.75 and \displaystyle \frac{12}{3.2} = 3.75.
(b) Her water is now in milliliters while Owen’s is in liters, so the second parts no longer match. The cross products are 7.5 \times 3200 = 24000 and 2 \times 12 = 24, which makes the statement false even though the mixtures are the same. Written with Owen’s water in milliliters too, 7.5 : 2000 = 12 : 3200 is true: 7.5 \times 3200 = 24000 = 2000 \times 12.
Is the table proportional?
Batches of pink paint are mixed from red and white paint.
| Red (L) | 2 | 4 | 5 | 7 | 10 |
|---|---|---|---|---|---|
| White (L) | 5 | 10 | 12.5 | 17 | 25 |
Are all the batches the same shade?
SolutionThe batches are the same shade exactly when every column is equivalent to every other, which is when white \div red is the same in every column:
Four batches have 2.5 L of white for each liter of red. The batch with 7 L of red does not: to match it would need 7 \times 2.5 = 17.5 L of white. So the table is not proportional. That batch has a little less white for each liter of red, so it is a slightly deeper pink.
Three parts at once
A concrete mix uses 1 bucket of cement, 2 of sand and 3 of gravel. Mateo mixes 4 buckets of cement, 8 of sand and 12 of gravel. Lena mixes 3 of cement, 6 of sand and 8 of gravel. Which of them has the right mix?
SolutionMateo’s amounts are each 4 times the mix: 4 \times 1 = 4, 4 \times 2 = 8 and 4 \times 3 = 12. One scale factor works for all three parts, so 1 : 2 : 3 = 4 : 8 : 12 and his mix is right.
Lena’s cement and sand are 3 times the mix, so the only possible scale factor is k = 3. Gravel would then be 3 \times 3 = 9 buckets, and she has 8. So 1 : 2 : 3 \ne 3 : 6 : 8. Notice that her cement and sand alone are in the right ratio, 1 : 2 = 3 : 6, which is why a three part ratio has to be checked with one factor for every part.
Adding one cup of each
Yusuf has a jug made from 3 cups of concentrate and 5 cups of water. He adds 1 more cup of concentrate and 1 more cup of water.
(a) Does the jug still taste the same? If not, is it stronger or weaker?
(b) After adding the cup of concentrate, how much water should he have added to keep the taste?
(c) For which ratios a : b does adding the same amount to both parts keep the ratio?
Solution(a) The new jug is 4 : 6. Test it against 3 : 5 with cross products:
They differ, so 3 : 5 \ne 4 : 6 and the taste has changed. Per cup of concentrate the water went from \displaystyle \frac{5}{3} \approx 1.67 cups to \displaystyle \frac{6}{4} = 1.5 cups, so the jug is now stronger.
(b) With 4 cups of concentrate, the water w must satisfy 3 : 5 = 4 : w. The scale factor from 3 to 4 is \displaystyle \frac{4}{3}, so
He started with 5 cups, so he should have added 6\tfrac{2}{3} - 5 = 1\tfrac{2}{3} cups of water.
(c) Only when a = b, as shown under Why it is true: the cross products of a : b and (a + n) : (b + n) are ab + an and ab + bn. In Yusuf’s jug a = 3 is less than b = 5, so an < bn and a(b + n) < b(a + n). Dividing both sides by the positive number b(b + n) gives \displaystyle \frac{a}{b} < \frac{a + n}{b + n}: after the top up there is more concentrate for each cup of water, which is why part (a) found the jug stronger.
Boundaries
- The fraction form needs nonzero second parts. \displaystyle \frac{a}{b} = \frac{c}{d} means nothing when b or d is 0. A ratio such as 3 : 0, concentrate with no water, can be written with a colon but not as a fraction, and the per one and lowest terms tests do not apply to it. A ratio 0 : 0 compares nothing and is not a ratio at all. Every test in this lesson assumes the four amounts are positive.
- The parts must line up. Both ratios must list the same quantities in the same order, and each matching pair must be in the same unit. Swap the order on one side, or measure water in liters on one side and milliliters on the other, and a true fact turns into a false statement (Examples 1 and 4).
- Scaling, not adding. Equivalent ratios come from multiplying both parts by one number. Adding the same amount to both parts keeps the ratio only when the two parts are equal (Example 7).
- The equation is about numbers; the taste is about the world. As an equation between numbers, \displaystyle \frac{a}{b} = \frac{c}{d} makes sense for any numbers with b and d nonzero, negatives included, and the cross product test still holds. Reading a proportion as “the same mixture” needs amounts, which are never negative, and it assumes the mixture is stirred evenly and that its taste or color depends only on how the parts compare.
Where this goes next
When three of the four numbers in a proportion are known, the fourth is fixed, and finding it is solving a proportion. The cross products from this lesson turn that into a one-step equation. Real problems arrive in sentences, and setting up a proportion from words is about putting each amount in its right place, the same matching of parts that Examples 1 and 4 insisted on.
A proportional table, where every row is equivalent to every other, has one number shared by all its rows: the per one amount. That number is the constant of proportionality, and it leads to the two other ways of seeing a proportional relationship, as a straight line through the origin and as an equation of the form y = kx. When two ratios are not proportional, comparing them says which is larger, the way Example 3 found the stronger jug. Much later, the sides of similar triangles are in proportion, and proportions in similar triangles uses exactly this idea to find lengths that cannot be measured.
Exercises
Reading and writing proportions
- Write this sentence as a proportion in fraction form, with cups of water on top: “5 cups of water for every 2 cups of concentrate is the same as 15 cups of water for every 6 cups of concentrate.” Is it true?
- Write 4 : 7 = 12 : 21 in words, and decide whether it is true.
- A paint chart says that 1 tin of yellow paint with 4 tins of white makes a cream color. (a) Write a proportion saying that 3 tins of yellow with 12 tins of white make the same cream. (b) Write the same fact as a proportion that compares yellow with yellow and white with white. Are both true?
Testing two ratios
Decide whether the two ratios are proportional, and show the test you used.
- 6 : 9 and 8 : 12
- 5 : 8 and 15 : 24
- 9 : 12 and 12 : 15
- 14 : 21 and 10 : 15
- 1.5 : 2 and 6 : 8
- 2.4 : 3 and 4 : 5.2
- \tfrac{3}{4} : 2 and 3 : 8
- 18 : 24 and 27 : 36
Mixtures and tables
- Jug 1 holds 2 cups of concentrate and 7 cups of water. Jug 2 holds 3 cups of concentrate and 10 cups of water. Do they taste the same? If not, which is stronger?
- A green is mixed from 3 L of blue paint and 2 L of yellow. A second can holds 7.5 L of blue and 5 L of yellow. Are the two greens the same shade?
- A bottle of plant food says to use 5 mL in 2 L of water. Arjun puts 12 mL into 5 L of water. Is his mixture the strength the bottle asks for? If not, is it stronger or weaker?
- A concrete mix is 1 part cement, 2 parts sand and 4 parts gravel. Which of these batches match it? (a) 3 : 6 : 12 (b) 2.5 : 5 : 10 (c) 4 : 8 : 14
- Is this table of plant food mixtures proportional? If not, which column breaks the pattern?
| Water (L) | 1 | 2 | 3 | 5 | 8 |
|---|---|---|---|---|---|
| Plant food (mL) | 4 | 8 | 12 | 20 | 30 |
- Is this table of lemonade mixtures proportional? If it is, how many cups of water go with each cup of concentrate?
| Concentrate (cups) | 4 | 6 | 10 | 12 |
|---|---|---|---|---|
| Water (cups) | 9 | 13.5 | 22.5 | 27 |
- Teo mixes 4 cups of orange concentrate with 10 cups of water. Later he pours in another 2 cups of concentrate and 5 cups of water. Does the jug taste the same as before? Explain why this is different from Example 7.
- Mika has 6 L of blue paint and 9 L of white. (a) She uses 1 L from each tin elsewhere and mixes the rest. Is it the shade that 6 L of blue with 9 L of white would have made? (b) Instead she mixes all 15 L and then pours out 3 L of the mixed paint. Is what is left the same shade? Find how much blue and white are left.
Reasoning
- Explain why 2 : 3 and 3 : 2 are not proportional, even though they use the same two numbers. For which numbers a and b is a : b = b : a?
- Two jugs taste the same: a : b = c : d. Show that pouring them together gives a jug that tastes the same as well, that is, (a + c) : (b + d) = a : b.
- Show that if \displaystyle \frac{a}{b} = \frac{c}{d}, with all four numbers positive, then \displaystyle \frac{a}{c} = \frac{b}{d}. What does the second proportion say about two jugs of lemonade?
- Show that if a : b = c : d and c : d = e : f, then a : b = e : f. What is the scale factor from a : b to e : f?
- The proportion a : b = c : d is true, all four numbers are positive, and a > b. Explain why c > d.
Answers
- \displaystyle \frac{5}{2} = \frac{15}{6}, and it is true. Both parts of 5 : 2 times 3 give 15 : 6; the cross products are 5 \times 6 = 30 and 2 \times 15 = 30.
- “4 is to 7 as 12 is to 21.” True: multiplying both parts of 4 : 7 by 3 gives 12 : 21, and the cross products are 4 \times 21 = 84 and 7 \times 12 = 84.
- (a) 1 : 4 = 3 : 12. (b) 1 : 3 = 4 : 12, yellow to yellow as white to white. Both are true: the scale factor is 3 in (a) and 4 in (b), and both have cross products 1 \times 12 = 12 and 4 \times 3 = 12.
- Proportional. Both simplify to \displaystyle \frac{2}{3}; cross products 6 \times 12 = 72 and 9 \times 8 = 72.
- Proportional. The scale factor is 3: 5 \times 3 = 15 and 8 \times 3 = 24.
- Not proportional. \displaystyle \frac{9}{12} = \frac{3}{4} but \displaystyle \frac{12}{15} = \frac{4}{5}; cross products 9 \times 15 = 135 and 12 \times 12 = 144.
- Proportional. Both simplify to \displaystyle \frac{2}{3}; cross products 14 \times 15 = 210 and 21 \times 10 = 210.
- Proportional. The scale factor is 4: 1.5 \times 4 = 6 and 2 \times 4 = 8.
- Not proportional. Cross products 2.4 \times 5.2 = 12.48 and 3 \times 4 = 12.
- Proportional. The scale factor is 4: \tfrac{3}{4} \times 4 = 3 and 2 \times 4 = 8; cross products \tfrac{3}{4} \times 8 = 6 and 2 \times 3 = 6.
- Proportional. Both simplify to \displaystyle \frac{3}{4}; the scale factor is 1.5, and the cross products are 18 \times 36 = 648 and 24 \times 27 = 648.
- No; jug 2 is stronger. Cross products 2 \times 10 = 20 and 7 \times 3 = 21 differ. Jug 1 has \displaystyle \frac{7}{2} = 3.5 cups of water per cup of concentrate, jug 2 only \displaystyle \frac{10}{3} \approx 3.33.
- Yes. Both parts of 3 : 2 times 2.5 give 7.5 : 5; cross products 3 \times 5 = 15 and 2 \times 7.5 = 15.
- No; it is slightly weaker. Cross products 5 \times 5 = 25 and 2 \times 12 = 24 differ. The bottle asks for \displaystyle \frac{5}{2} = 2.5 mL per liter, and Arjun’s is \displaystyle \frac{12}{5} = 2.4 mL per liter.
- (a) and (b) match; (c) does not. (a) is the mix times 3 and (b) the mix times 2.5. In (c) cement and sand give a factor of 4, so gravel would have to be 4 \times 4 = 16, not 14.
- Not proportional; the column with 8 L breaks the pattern. The first four columns have 4 mL per liter, but \displaystyle \frac{30}{8} = 3.75. To match it would need 8 \times 4 = 32 mL.
- Proportional, with 2.25 cups of water per cup of concentrate: \displaystyle \frac{9}{4} = \frac{13.5}{6} = \frac{22.5}{10} = \frac{27}{12} = 2.25.
- Yes. The jug is now 6 : 15, which simplifies to \displaystyle \frac{2}{5}, the same as \displaystyle \frac{4}{10}. What he poured in, 2 : 5, is itself a batch of the same taste, half of the first jug. In Example 7 the added 1 : 1 was a different mixture.
- (a) No. She mixes 5 : 8, and the cross products with 6 : 9 are 5 \times 9 = 45 and 8 \times 6 = 48. (b) Yes, with 4.8 L of blue and 7.2 L of white left. Every liter of the mixed paint holds \displaystyle \frac{6}{15} = 0.4 L of blue and 0.6 L of white, so 3 L removes 1.2 L of blue and 1.8 L of white, and 4.8 : 7.2 simplifies to \displaystyle \frac{2}{3}.
- 2 : 3 has less of the first thing than the second and 3 : 2 has more, so they describe different mixtures; the cross products are 2 \times 2 = 4 and 3 \times 3 = 9. In general the cross products of a : b and b : a are a^2 and b^2, which for positive numbers are equal only when a = b.
- Write c = ka and d = kb. Then a + c = (1 + k)a and b + d = (1 + k)b, so the poured together jug is a : b with both parts multiplied by 1 + k, and equivalent to it.
- Both proportions are the same equation in disguise: the cross products of \displaystyle \frac{a}{b} = \frac{c}{d} give ad = bc, and those of \displaystyle \frac{a}{c} = \frac{b}{d} give ad = cb. About two jugs: the concentrate in the first compares with the concentrate in the second the same way the water does. That shared comparison is the scale factor.
- Write c = ka, d = kb and e = mc, f = md. Then e = mka and f = mkb, so one number, mk, scales a : b to e : f. The scale factors multiply.
- There is a positive k with c = ka and d = kb. Multiplying both sides of a > b by the positive number k keeps the inequality, so ka > kb, that is, c > d. A jug with more concentrate than water stays that way however many copies are made.