Math 6 · 6.RP.A.3 · 6.RP.A.1

Scaling a ratio up or down

A jug of cordial tastes just right. How do you make a bigger jug, or a smaller one, that tastes exactly the same?

You need

Simplifying a fraction to simplest formGrade 4 What a ratio is: language and notationMath 6

Unlocks

Finding a missing value in a pair of equal ratiosMath 6 Rates and unit ratesMath 6

Where you are 5 in Equivalent ratios and ratio tables

Fractions, ratio and percent3

Ratio6

Equivalent ratios and ratio tables5

Tape diagrams and sharing in a ratio The double number line

Scaling a ratio up or down

Finding a missing value in a pair of equal ratios Ratio tables

When you reach for it

A ratio such as 3 : 7 describes a mix: for every 3 scoops of cordial there are 7 scoops of water. (How a ratio is written and read aloud is in what a ratio is.) A recipe comes in one size, but jugs, bags of feed and wheelbarrows come in many. You reach for scaling whenever you need more or less of a mix that has to stay the same mix: a party bowl of cordial that tastes like the glass, a big batch of mortar as strong as the small one, a week of guinea pig food in the same balance as a day’s.

The same move answers two more questions. Are two mixes really the same, even though their numbers are different? And what are the smallest whole numbers that describe a mix, so it is easy to read and compare? The first question is about equivalent ratios, the second about simplest form. Every ratio in this lesson compares one part of a mix with another part, a Part-to-part and part-to-whole ratios.

The procedure

Scaling up: copies of the group

Read 3 : 7 as a group: 3 scoops of cordial that go with 7 scoops of water. Two batches are two copies of that group, so 3 + 3 = 6 scoops of cordial and 7 + 7 = 14 scoops of water, the ratio 6 : 14. Three batches are three copies, 9 : 21.

Figure 1Three batches of 3 : 7 make 9 : 21. Every scoop of cordial still sits in a group with its own 7 scoops of water.

Every scoop of cordial in the big bowl still belongs to a group of 3 that has its own 7 scoops of water, so the bowl tastes the same as one glass. Making copies is multiplying, so

3 \times 3 : 7 \times 3 = 9 : 21.

Both amounts are multiplied by the same number, here 3, called the scale factor. Multiplying only one amount breaks the groups apart: 9 scoops of cordial with 7 of water is three groups of cordial sharing one group’s water, a far stronger drink.

Two ratios are equivalent when one is made from the other by multiplying both amounts by the same number, any number except 0. So 3 : 7, 6 : 14 and 9 : 21 are equivalent: they are the same mix in different amounts.

Scaling down: dividing both amounts

Scaling down runs the same move backwards. Dividing both amounts of 9 : 21 by 3 gives 9 \div 3 : 21 \div 3 = 3 : 7, one batch out of three. Dividing by 3 is the same as multiplying by \displaystyle \frac{1}{3}, so scaling down gives an equivalent ratio for exactly the same reason scaling up does.

The scale factor need not be a whole number. A guinea pig recipe of 6 handfuls of hay to 5 scoops of pellets, halved, is 3 : 2.5. It is still the same mix, because each amount was cut in half, and half a scoop is a perfectly good amount.

This is the move you already know from fractions: \displaystyle \frac{9}{21} and \displaystyle \frac{3}{7} are Equivalent fractions because top and bottom were divided by the same number, and 9 : 21 and 3 : 7 are equivalent ratios for the same reason.

Simplest form

A ratio of whole numbers is in simplest form when no whole number bigger than 1 divides both amounts. 3 : 7 is in simplest form. 9 : 21 is not, because 3 divides both.

To reach simplest form, divide both amounts by a number that divides them both, and repeat while such a number is left, just as in simplifying a fraction. Dividing by the greatest common factor finishes in one step. Each division gives an equivalent ratio, so the simplest form is the same mix in the smallest whole numbers that can describe it.

That gives a test for whether two whole-number ratios are equivalent: bring both to simplest form and see whether they match.

Why adding the same amount changes the mix

Scaling multiplies. Adding the same amount to both parts looks like another way to make more, but it changes the mix. Take a weak cordial, 1 : 3. Two batches make 2 : 6. Adding one scoop of each instead makes 2 : 4.

Figure 2Two batches of 1 : 3 give 2 : 6. Adding one scoop of each instead gives 2 : 4, the same cordial with less water.

Both have 2 scoops of cordial, but 2 : 4 has 4 scoops of water where two batches have 6. The extra scoop of cordial arrived without its own 3 scoops of water, so the drink is stronger. Dividing by 2 shows 2 : 4 is 1 : 2, a different mix from 1 : 3.

Adding keeps a mix only when what is added is itself in the same ratio: adding another 1 : 3 batch is one more copy of the group. The one ratio that survives adding equal amounts is 1 : 1, because equal amounts with equal amounts added stay equal.

Worked examples

Example 1

A party bowl

Greta’s cordial is 3 scoops of cordial to 7 scoops of water. She wants a bowl holding 5 times as much. How many scoops of each does she need, and how many scoops of drink is that?

Solution

Five times as much is five copies of the group, so both amounts are multiplied by the scale factor 5:

3 \times 5 : 7 \times 5 = 15 : 35.

She needs 15 scoops of cordial and 35 scoops of water. One batch is 3 + 7 = 10 scoops of drink, so five batches are 50 scoops. As a check, 15 + 35 = 50.

Example 2

A small batch of mortar

Sven’s mortar uses 20 buckets of sand and 5 buckets of cement. For a small repair he needs a fifth of the batch, and for one loose brick a tenth of it. How much of each goes into these?

Solution

A fifth means dividing both amounts by 5: 20 \div 5 : 5 \div 5 = 4 : 1, so 4 buckets of sand and 1 of cement. Because 4 and 1 share no factor bigger than 1, this is also the simplest form of 20 : 5.

A tenth means dividing both by 10: 20 \div 10 = 2 and 5 \div 10 = 0.5, giving 2 : 0.5, so 2 buckets of sand and half a bucket of cement. It is the same mortar: doubling 2 : 0.5 gives back 4 : 1.

Example 3

Simplest form by grouping

Over one week Obi’s guinea pigs eat 18 scoops of pellets and 24 handfuls of hay. Write the ratio of pellets to hay in simplest form and say what it means.

Solution

The ratio is 18 : 24. Both numbers are even, so divide both by 2 to get 9 : 12. Both of those divide by 3, giving 3 : 4. No whole number bigger than 1 divides both 3 and 4, so 3 : 4 is the simplest form.

The two steps divided by 2 and then by 3, which is dividing by 6 in all, and 6 is the greatest common factor of 18 and 24: one division by 6 gives 3 : 4 at once. The picture shows why. The counters sort into 6 equal groups, each of 3 scoops of pellets and 4 handfuls of hay.

Figure 318 scoops of pellets and 24 handfuls of hay sort into 6 equal groups of 3 and 4, so 18 : 24 is 3 : 4.

So for every 3 scoops of pellets the guinea pigs eat 4 handfuls of hay.

Example 4

Are two feeds the same mix?

Over a month Lucia’s guinea pigs eat 45 scoops of pellets and 60 handfuls of hay. Is that the same mix as Obi’s 18 : 24?

Solution

Bring 45 : 60 to simplest form. Both end in 0 or 5, so divide by 5: 9 : 12. Both divide by 3: 3 : 4. (The greatest common factor is 15, and 45 \div 15 = 3, 60 \div 15 = 4 in one step.)

Obi’s 18 : 24 also simplifies to 3 : 4, so the two ratios are equivalent: the same mix. Lucia’s numbers are bigger because they cover more days, not because her guinea pigs get a different balance.

Example 5

Scaling by a fraction

Rohan feeds his guinea pigs 6 handfuls of hay to 5 scoops of pellets for one day. He needs half a day’s feed, and one and a half days’ feed. Find the amounts, and say whether 6 : 5 is in simplest form.

Solution

Half a day’s feed scales both amounts by \displaystyle \frac{1}{2}: \displaystyle 6 \times \frac{1}{2} = 3 and \displaystyle 5 \times \frac{1}{2} = 2.5, so 3 : 2.5.

One and a half days scales both by 1.5: 6 \times 1.5 = 9 and 5 \times 1.5 = 7.5, so 9 : 7.5. As a check, one and a half days is three half days, and 3 \times 3 = 9, 2.5 \times 3 = 7.5.

6 = 2 \times 3 and 5 has no factor but 1 and itself, so no whole number bigger than 1 divides both, and 6 : 5 is in simplest form. 3 : 2.5 has smaller numbers, but simplest form is about whole numbers: doubling 3 : 2.5 brings back 6 : 5.

Example 6

Adding two scoops of each

A cordial is mixed 3 : 7. One suggestion for making more is to add 2 scoops of cordial and 2 of water to a batch, which gives 5 : 9. Does 5 : 9 taste the same as 3 : 7? If not, which is stronger?

Solution

Compare the two mixes with the same amount of cordial. 15 is a multiple of both 3 and 5. Scaling 3 : 7 by 5 gives 15 : 35, and scaling 5 : 9 by 3 gives 15 : 27.

With 15 scoops of cordial, the 3 : 7 mix has 35 scoops of water and the 5 : 9 mix only 27. Same cordial, less water, so 5 : 9 is stronger, and the two ratios are not equivalent. Both are already in simplest form and they differ, which says the same thing.

Example 7

Topping up a bowl

Amara has a bowl of 9 scoops of cordial and 21 scoops of water. Show that adding one more 3 : 7 batch keeps the taste, but adding 3 scoops of each does not, and say which way the taste moves.

Solution

Dividing both amounts by 3 shows 9 : 21 is 3 : 7: the bowl is three batches. Adding one more batch gives 12 : 28, and dividing both by 4 gives 3 : 7 again, so the taste is unchanged: it is four batches.

Adding 3 scoops of each gives 12 : 24, and dividing both by 12 gives 1 : 2, which is not 3 : 7. With 12 scoops of cordial the true mix is 12 : 28, so the bowl now has 4 scoops too little water and tastes stronger. Whatever is added has to be in the ratio 3 : 7 itself.

Boundaries

The scale factor is never 0. Multiplying both amounts by 0 gives 0 : 0, which is no mix at all, and it cannot be undone because nothing can be divided by 0.
Only multiplying or dividing both amounts by the same number keeps a ratio. Adding or subtracting the same amount changes every ratio except 1 : 1.
Simplest form is for whole numbers. A ratio such as 3 : 2.5 is first scaled to whole numbers (6 : 5) before asking whether it is in simplest form.
Order stays fixed. Scaling 3 : 7 cordial to water gives 6 : 14 cordial to water, never 14 : 6.
Both amounts keep their units. A 3 : 7 mix measured in teaspoons and a 3 : 7 mix measured in cups taste the same, but measuring the cordial in teaspoons and the water in cups is a different mix.

Where this goes next

Scaling is the engine behind the rest of Equivalent ratios and ratio tables. When one amount of a new mix is known, the scale factor can be worked out from it and applied to the other amount, which is finding a missing value. Writing equivalent ratios in rows gives a ratio table, and marking them on two lines side by side gives a The double number line. Scaling a ratio so that one amount becomes exactly 1 gives a Rates and unit rates: 3 : 7 scaled by \displaystyle \frac{1}{3} is \displaystyle 1 : 2\frac{1}{3}, two and a third scoops of water for every scoop of cordial.

Exercises

Scaling up and down

Scale the ratio 3 : 7 by a factor of 4.
Scale the ratio 6 : 5 by a factor of 3.
Scale the ratio 40 : 16 down by dividing both amounts by 8.
Sven’s mortar for a garden wall uses 8 buckets of sand and 2 buckets of cement. (a) How much of each is needed for a wall that takes three times as much mortar? (b) Write 8 : 2 in simplest form.
Lucia mixes 5 scoops of cordial with 12 scoops of water. How much of each goes into half a jug of the same drink?
Rohan’s hay to pellets recipe is 6 : 5. Find the amounts for a third of a batch.

Equivalent or not

Is 9 : 21 equivalent to 3 : 7?
Is 12 : 10 equivalent to 6 : 5?
Is 7 : 6 equivalent to 6 : 5?
Are 15 : 35 and 6 : 14 equivalent?

Simplest form

Write 12 : 20 in simplest form.
Write 36 : 48 in simplest form.
Write 28 : 42 in simplest form.
Write 50 : 15 in simplest form.
Write 16 : 27 in simplest form.
A mortar mix uses 75 shovels of sand to 25 shovels of cement. Write the ratio of sand to cement in simplest form.
Obi’s feed 18 : 24 and Lucia’s feed 45 : 60 both simplify to 3 : 4. Find the one number that both amounts of 18 : 24 can be multiplied by to give 45 : 60.

Adding the same amount to both

Amara’s cordial is 2 : 5, cordial to water. Adding 3 scoops of each gives 5 : 8. Does it taste the same? If not, is it stronger or weaker?
Sven’s mortar is 4 : 1, sand to cement. Adding 2 buckets of each gives 6 : 3. Write 6 : 3 in simplest form. Does the new mix have more or less sand for each bucket of cement?

Explaining why

Use groups of counters to explain why multiplying both amounts of 3 : 7 by 4 gives the same mix, and why multiplying only the cordial by 4 does not.
Explain why adding the same amount to both parts of 1 : 1 never changes it, but adding the same amount to both parts of 3 : 7 always does.
Explain why 0 is never used as a scale factor.
Two whole-number ratios have the same simplest form. Explain why they must be equivalent to each other.

Answers

12 : 28. Both amounts times 4: 3 \times 4 = 12 and 7 \times 4 = 28.
18 : 15. Both amounts times 3: 6 \times 3 = 18 and 5 \times 3 = 15.
5 : 2. 40 \div 8 = 5 and 16 \div 8 = 2.
(a) 24 : 6, that is 24 buckets of sand and 6 of cement, from 8 \times 3 and 2 \times 3. (b) 4 : 1, dividing both by 2.
2.5 : 6, that is 2.5 scoops of cordial and 6 of water: half of 5 is 2.5 and half of 12 is 6.
\displaystyle 2 : 1\frac{2}{3}, that is 2 handfuls of hay and \displaystyle 1\frac{2}{3} scoops of pellets: 6 \div 3 = 2 and \displaystyle 5 \div 3 = \frac{5}{3} = 1\frac{2}{3}.
Yes. Multiplying both amounts of 3 : 7 by 3 gives 9 : 21.
Yes. Multiplying both amounts of 6 : 5 by 2 gives 12 : 10.
No. Make the first amounts match: 6 : 5 times 7 is 42 : 35, and 7 : 6 times 6 is 42 : 36. Same first amount, different second amount, so the mixes differ.
Yes. 15 : 35 divided by 5 is 3 : 7, and 6 : 14 divided by 2 is 3 : 7. Same simplest form.
3 : 5. The greatest common factor is 4: 12 \div 4 = 3, 20 \div 4 = 5.
3 : 4. The greatest common factor is 12: 36 \div 12 = 3, 48 \div 12 = 4.
2 : 3. The greatest common factor is 14: 28 \div 14 = 2, 42 \div 14 = 3.
10 : 3. The greatest common factor is 5: 50 \div 5 = 10, 15 \div 5 = 3.
16 : 27 is already in simplest form. 16 = 2 \times 2 \times 2 \times 2 and 27 = 3 \times 3 \times 3 share no factor bigger than 1.
3 : 1. Dividing both by 25: 75 \div 25 = 3, 25 \div 25 = 1.
2.5. 18 : 24 is 6 groups of 3 : 4 and 45 : 60 is 15 groups, so the factor is 15 \div 6 = 2.5. Check: 18 \times 2.5 = 45 and 24 \times 2.5 = 60.
Not the same; 5 : 8 is stronger. With 10 scoops of cordial, 2 : 5 scaled by 5 is 10 : 25, while 5 : 8 scaled by 2 is 10 : 16: same cordial, less water.
2 : 1, dividing both by 3. The new mix has less sand for each bucket of cement: 2 buckets instead of 4.
3 : 7 is one group of 3 cordial counters with 7 water counters. Multiplying both amounts by 4 makes 4 copies of the group, 12 : 28, and every scoop of cordial still sits in a group with its own 7 scoops of water, so every sip is the same mix. Multiplying only the cordial gives 12 : 7: four groups’ worth of cordial sharing one group’s water, a much stronger drink.
In 1 : 1 the two amounts are equal, and adding the same amount to two equal amounts leaves them equal, so the mix is still 1 : 1 (for example 4 : 4 becomes 7 : 7). A mix equivalent to 3 : 7 is made of groups with 7 scoops of water for every 3 of cordial, so going from one such mix to a bigger one adds more water than cordial. Equal added amounts bring too little water, so the mix always ends up stronger.
Multiplying both amounts by 0 gives 0 : 0: no cordial and no water, so nothing is left to taste. It would also turn every ratio into 0 : 0, making 3 : 7 and 1 : 1 look like the same mix, and there is no way back, because nothing can be divided by 0.
Each ratio is its simplest form scaled up by some whole number. Dividing both amounts of the first ratio by its number takes it back to the shared simplest form, and multiplying both amounts by the second ratio’s number takes that to the second ratio. For example 12 : 20 is 3 : 5 times 4, and 9 : 15 is 3 : 5 times 3. Dividing both amounts of the first by 4 and then multiplying both by 3 turns it into the second, and both steps do the same thing to both amounts. Together they multiply both amounts by \displaystyle \frac{3}{4}: \displaystyle 12 \times \frac{3}{4} = 9 and \displaystyle 20 \times \frac{3}{4} = 15. So the two ratios are equivalent.