Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Year 7, Unit 1, lesson 5) What is the highest common factor of 40 and 64?
Answer
8, the largest number that divides both 40 and 64. - (Year 7, Unit 5, lesson 1) Fill the gap: 2/3 = ?/15.
Answer
10, because 3 × 5 = 15 and 2 × 5 = 10. - (Lesson 1) A bag has 5 red and 7 blue counters. Write the ratio of red counters to all the counters.
Answer
5 : 12, because the whole is 5 + 7 = 12.
Words used in this lesson
- equal ratios
- ratios that say the same thing in different numbers. For example: 4 : 12 and 1 : 3.
- simplest form
- the ratio with the smallest whole numbers. For example: 2 : 3, not 4 : 6.
- common factor
- a number that divides both parts exactly. For example: 6 divides 12 and 18.
- highest common factor
- the largest number that divides both parts exactly. For example: 6 for 12 and 18.
- same unit
- both amounts measured in the same way. For example: 500 cm and 80 cm.
The same comparison, in smaller numbers
A squash bottle says: mix 4 cups of concentrate with 12 cups of water. A second bottle says: mix 1 cup with 3 cups. A third says: 2 cups with 6 cups. Are these three different drinks, or one drink written three ways?
Predict first: by what number would you divide 4 : 12 to get 1 : 3, and what must you do to both numbers? Then explore: cut the two bars into equal groups and count the groups.
20 and 24 both split into groups of 4: 20 is 5 groups and 24 is 6 groups, so 20 : 24 simplifies to 5 : 6.
Show me the steps
- 20 and 24: the biggest group that fits both is 4. 20 is 5 groups and 24 is 6 groups, so 20 : 24 = 5 : 6.
- 9 and 15: the biggest group is 3. 9 is 3 groups and 15 is 5 groups, so 9 : 15 = 3 : 5.
- 7 and 13: the only group that fits both is 1, so 7 : 13 is already in its simplest form.
- 16 and 16: the group is 16, one group each, so 16 : 16 = 1 : 1.
- 6 and 24: the biggest group is 6. 6 is 1 group and 24 is 4 groups, so 6 : 24 = 1 : 4.
Equal ratios say the same thing in different numbers. Divide both parts by the same number and the ratio is equal: 4 : 12 = 2 : 6 = 1 : 3. The same is true of multiplying both parts.
The simplest form uses the highest common factor: the biggest group that fits both parts. The number of groups on each side is the simplest form, so 4 : 12 is 1 group to 3 groups.
Units first, then simplify. Write both amounts in one unit. A ratio can also be written 1 : n by dividing both parts by the first number, and n can be a decimal: 4 : 5 = 1 : 1.25.
Back to the bottles: divide 4 : 12 by 4 and it is 1 : 3. Divide 2 : 6 by 2 and it is 1 : 3 as well. All three bottles make the same drink, written in three sets of numbers.
A common mistake: “5 m to 80 cm is 5 : 80, which is 1 : 16.” Different units cannot be compared as they stand. 5 m is 500 cm, so the ratio is 500 : 80, which is 25 : 4.
Worked example: simplify with the highest common factor
Write 36 : 48 in its simplest form.
- List the factors. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The highest common factor is 12.
- Divide both parts by 12: 36 ÷ 12 = 3 and 48 ÷ 12 = 4.
- Check: 3 × 12 = 36 and 4 × 12 = 48. 3 : 4.
A rope is 2 m long and a ribbon is 80 cm long. Write rope to ribbon in simplest form.
- Same unit first: 2 m = 200 cm, so the ratio is 200 : 80.
- The highest common factor of 200 and 80 is 40.
- 200 ÷ 40 = 5 and 80 ÷ 40 = 2. 5 : 2.
Your go. The first step is given. A ribbon of 90 cm is cut from a roll that is 3 m long. Write the ratio of ribbon to roll in simplest form.
- Same unit first: 3 m = 300 cm, so the ratio is 90 : 300.
- Now find the highest common factor and divide, then check.
Check the answer
The highest common factor of 90 and 300 is 30. 90 ÷ 30 = 3 and 300 ÷ 30 = 10, so the ratio is 3 : 10. Check: 3 × 30 = 90 and 10 × 30 = 300.
Check before you go on
1. Which is 24 : 36 in its simplest form?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Fill the gap so that the ratios are equal: 15 : 25 = 3 : ?
3. Which pair of ratios is equal?
Level 2 · Building
4. Write 56 : 42 in its simplest form. What is the first number?
5. A path is 4 m long and a hedge is 250 cm long. Write path to hedge in simplest form. What is the second number?
6. Write 2 : 5 in the form 1 : n. What is n? Give n as a decimal.
Level 3 · Stretch
7. Write 12 : 18 : 30 in its simplest form. What is the last number?
8. A ratio in simplest form is 4 : 7. Before it was simplified, the first number was 36. What was the second number?
Something to think about
Is there a ratio that cannot be simplified at all? Find three, then try to say a rule for when a : b is already in its simplest form.
One way to think about it
1 : 5, 3 : 4 and 7 : 9 cannot be simplified. A ratio is in simplest form when its two numbers have no common factor above 1. Any ratio with a 1 in it fits, and so does any pair of neighbouring whole numbers such as 8 : 9, because a number and the next one share no factor above 1.
Exit ticket
9. Ravi simplifies 12 : 18 by taking 6 off both numbers and writes 6 : 12. What is wrong?