Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Year 7, Unit 5, lesson 1) Write 6/8 in its simplest form.
Answer
3/4. Divide the top and bottom by 2. - (Year 7, Unit 2, lesson 1) What fraction is 25%?
Answer
1/4, because 25 out of 100 is 25/100 = 1/4. - (Year 7, Unit 1, lesson 5) What is the highest common factor of 12 and 18?
Answer
6, because 6 is the largest number that divides both.
Words used in this lesson
- ratio
- a comparison of two amounts, written with a colon. For example: red to blue is 3 : 5.
- part
- one of the amounts being compared. For example: the 3 in 3 : 5.
- whole
- all the parts added together. For example: 3 + 5 = 8 beads in all.
- order
- which amount is named first. For example: limes to mangoes is not mangoes to limes.
- same unit
- both amounts measured in the same way. For example: 300 cm and 40 cm, not 3 m and 40 cm.
For every one of these, so many of those
A bag holds 3 red beads and 5 blue beads. You can say “3 red and 5 blue”, or you can say that the ratio of red to blue is 3 : 5. A ratio compares amounts by saying how many of one there are for every so many of the other. It does not need the whole bag: 6 red and 10 blue is the same comparison.
Predict first: in that bag, what is the ratio of blue to red? And how many of the beads are red, out of how many in all? Then build your own bag with the sliders and test your idea.
2 counters in the first group and 5 in the second: first to second is 2 : 5. Swap the order and it is 5 : 2. The first group is 2 out of 7 counters in all.
Show me the steps
- 2 counters in the first group and 5 in the second: first to second is 2 : 5, and second to first is 5 : 2. Swapping the order swaps the numbers.
- 4 and 4: the ratio is 4 : 4. The groups are equal, and the first group is 4 out of 8, which is half of all the counters.
- 6 and 3: first to second is 6 : 3, and the first group is 6 out of 9.
- 1 and 8: first to second is 1 : 8, but the first group is 1 out of 9, not 1 out of 8. The whole is the two parts added together.
A ratio compares two amounts. a : b says: for every a of the first thing there are b of the second. The order matters: 2 : 5 and 5 : 2 are different ratios, and the order of the numbers follows the order of the words.
A ratio of one group to another compares part with part. A ratio can also compare a part with the whole, and the whole is all the parts added: in 2 : 5 the first group is 2 out of 2 + 5 = 7, so first to whole is 2 : 7.
Same unit first. When both amounts are the same kind, put them in one unit before you write the ratio: 1 kg and 250 g is 1000 g : 250 g.
Back to the bag: blue to red is 5 : 3. The bag holds 3 + 5 = 8 beads, so 3 out of 8 are red. It is not 3 out of 5, because the 5 counts only the blue beads.
A common mistake: “The ratio of altos to sopranos is 3 : 8, so 3/8 of the choir are altos.” The 8 counts only the sopranos. The choir is 3 + 8 = 11 singers, so the altos are 3 out of 11.
Worked example: writing the ratio in the order asked
A tray holds 12 jam tarts and 20 lemon tarts. Write the ratio of jam to lemon, and the ratio of lemon to all the tarts.
- Jam is named first, so jam is the first number: jam to lemon is 12 : 20.
- All the tarts: 12 + 20 = 32.
- Lemon is named first this time: lemon to all is 20 : 32.
Your go. The first step is given. A box holds 9 pens that write blue and 6 that write black. Write the ratio of black pens to blue pens.
- Black is named first, so the first number counts the black pens: 6.
- Now write the second number yourself, then check.
Check the answer
Blue is second, so the ratio is 6 : 9. Writing 9 : 6 would be the ratio of blue to black, the other order.
Check before you go on
1. A fruit stall has 7 mangoes and 4 limes. Which ratio compares the limes to the mangoes?
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. The ratio of boys to girls in a swimming club is 5 : 7. How many parts make up the whole club?
3. A dish has rice and lentils in the ratio 2 : 9 (rice to lentils). Which sentence says the same thing?
Level 2 · Building
4. A bag has 7 red, 2 green and 4 yellow sweets. The ratio of red sweets to all the sweets is 7 : ?. What is the missing number?
5. Write the ratio of a 3 m ribbon to a 40 cm ribbon with both lengths in centimetres. What is the first number?
6. In class 8B, 14 pupils walk to school, 9 cycle and 7 come by bus. Write the ratio of pupils who walk to pupils who do not walk. What is the second number?
Level 3 · Stretch
7. A bag has red and blue beads in the ratio 1 : 3. What percentage of the beads are blue?
8. A shelf holds comics, novels and atlases in the ratio 2 : 3 : 7. The ratio of atlases to the books that are not atlases is 7 : ?. What is the missing number?
Something to think about
A club has boys and girls in the ratio 1 : 1. Does that tell you how many students are in the club? Write down three different clubs that fit.
One way to think about it
No. 1 : 1 only says the two groups are the same size. Any equal numbers fit: 5 boys and 5 girls (10 students), 12 and 12 (24), 30 and 30 (60). A ratio gives the shape of the comparison, not the size of the club.
Exit ticket
9. A pot has 4 cups of rice and 6 cups of water, so rice to water is 4 : 6. Which statement is true?