Dividing in a Given Ratio

Worked example

1. 1

   $1.5\text{ kg}:600\text{ g}$ — The question. The units are different, so you cannot compare yet.

2. 2

   $1500:600$ — Change kg to g: 1.5 × 1000 = 1500. Now the units match, so drop them.

3. 3

   $5:2$ — Divide both by 300. 5 and 2 share no factor, so you are done.

Worked example

1. 1

   $2+3+7=12\text{ parts}$ — Add the ratio numbers. All 12 parts are the same size, and together they make 360.

2. 2

   $360÷12=30$ — Share 360 equally between the 12 parts, so one part is 30.

3. 3

   $2\times 30=60$ — The smallest share has 2 parts, so it is 2 lots of 30.

4. 4

   $7\times 30=210$ — The largest share has 7 parts, so it is 7 lots of 30.

5. 5

   $210-60=150$ — Subtract to find how much more the largest share gets.

6. 6

   $60+90+210=360✓$ — Check: the middle share is 3 × 30 = 90, and all three add back to 360.

Worked example

1. 1

   $8-3=5\text{ parts}$ — Ben has 8 parts and Ali has 3, so Ben has 5 parts more.

2. 2

   $5\text{ parts}=\pounds 35$ — Those 5 extra parts are the £35 extra that Ben gets.

3. 3

   $\text{one part}=35÷5=\pounds 7$ — Divide by 5 to find one part.

4. 4

   $3+8=11\text{ parts}$ — Together they share 11 parts.

5. 5

   $11\times 7=\pounds 77$ — 11 parts at £7 each is the whole amount shared.

6. 6

   $8\times 7-3\times 7=56-21=35✓$ — Check: Ben gets £56 and Ali £21, and the gap is £35.

Worked example

1. 1

   $5+7=12\text{ parts}$ — Girls and boys together make 12 parts.

2. 2

   $96÷12=8$ — Share 96 students equally between 12 parts, so one part is 8 students.

3. 3

   $\text{girls }5\times 8=40$ — Girls have 5 parts. Find the people before you take any fraction.

4. 4

   $\text{boys }7\times 8=56$ — Boys have 7 parts. Check: 40 + 56 = 96.

5. 5

   $\frac{3}{5}\times 40=24$ — The 3/5 is of the girls only: 40 ÷ 5 = 8, then 3 × 8 = 24 girls walk.

6. 6

   $\frac{1}{4}\times 56=14$ — The 1/4 is of the boys only: 56 ÷ 4 = 14 boys walk.

7. 7

   $24+14=38$ — Add them to get everyone who walks.

8. 8

   $\frac{38}{96}$ — The question asks for a fraction of all students, so put the walkers over all 96.

9. 9

   $=\frac{19}{48}$ — Divide top and bottom by 2. 19 is prime and does not divide 48, so it is fully simplified.

Worked example

1. 1

   $3:2=6:4$ — Dogs are 2 in the first ratio and 4 in the second. Double the first ratio so dogs are 4 in both.

2. 2

   $\text{cats : dogs : rabbits}=6:4:5$ — Now the dog parts match, so all the parts are the same size and you can join the ratios.

3. 3

   $5\text{ parts}=30$ — Rabbits are 5 parts, and there are 30 rabbits.

4. 4

   $\text{one part}=30÷5=6$ — Divide by 5, so one part is 6 animals.

5. 5

   $\text{cats}=6\times 6=36$ — Cats are 6 parts, so there are 36 cats.

6. 6

   $\text{dogs}=4\times 6=24$ — Check: 36 : 24 is 3 : 2, and 24 : 30 is 4 : 5. Both match the question.

Worked example

1. 1

   $\frac{3}{8}\times 720=270$ — The fraction is of the whole 720, so work it out first: 720 ÷ 8 = 90, and 3 × 90 = 270 kg sold.

2. 2

   $720-270=450$ — Take it away. Only the 450 kg left is shared.

3. 3

   $2+7=9\text{ parts}$ — Add the ratio numbers to get the parts.

4. 4

   $450÷9=50$ — Share the 450, not the 720. One part is 50 kg.

5. 5

   $7-2=5\text{ parts}$ — The second shop has 5 more parts than the first.

6. 6

   $5\times 50=250\text{ kg}$ — 5 parts at 50 kg each is how much more the second shop gets.

7. 7

   $100+350=450✓$ — Check: the shares are 2 × 50 = 100 and 7 × 50 = 350, and they add to the 450 left.

Worked example

1. 1

   $2+3+4=9\text{ parts}$ — Add the ratio numbers.

2. 2

   $180÷9=20$ — Share the £180 equally between 9 parts, so one part is £20.

3. 3

   $\text{Ann }40,\text{Bo }60,\text{Cy }80$ — 2, 3 and 4 parts, each times 20. These are the amounts before anyone gives money.

4. 4

   $\text{Ann }40+15=55$ — Ann receives £15, so add it.

5. 5

   $\text{Cy }80-15=65$ — Cy gives the same £15 away, so take it off. Bo does not change.

6. 6

   $55:60:65$ — The new amounts, in the order Ann : Bo : Cy. They still add to 180.

7. 7

   $11:12:13$ — Divide all three by 5. 11, 12 and 13 share no factor, so this is the simplest form.

Worked example

1. 1

   $16-4=12$ — Go back 4 years. Sam was 12 when the ratio was true.

2. 2

   $12÷3=4$ — Sam had 3 parts, so one part is 4 years.

3. 3

   $5\times 4=20$ — Lee had 5 parts, so Lee was 20 then.

4. 4

   $20+4=24$ — Add the 4 years back. Lee is 24 now.

Worked example

1. 1

   $3\times 10=30\text{c}$ — One set has 3 coins of 10c. Their value is 30c.

2. 2

   $2\times 50=100\text{c}$ — The same set has 2 coins of 50c, worth 100c.

3. 3

   $30+100=130\text{c}$ — So one set of 5 coins is worth 130c.

4. 4

   $\$7.80=780\text{c}$ — Change dollars to cents so both amounts use the same unit.

5. 5

   $780÷130=6$ — The jar holds 6 sets of coins.

6. 6

   $3\times 6=18,2\times 6=12$ — 18 coins of 10c and 12 coins of 50c, still in the ratio 3 : 2.

7. 7

   $18+12=30\text{ coins}$ — Add them. Check: 180c + 600c = 780c.