Inverse Proportion in Real Life

Worked example

1. 1

   $6\times 10=60$ — The whole job: 6 workers for 10 days is 60 days of work. Multiply first, because the size of the job never changes.

2. 2

   $60÷4=15\text{ days}$ — Now 4 workers share the same 60 days of work, so divide by 4. Fewer workers, so more days: 15 is more than 10.

Worked example

1. 1

   $60÷5=12\text{ workers}$ — The job is still 60 days of work. To fit it into 5 days, divide by 5 to get the number of workers.

2. 2

   $12\times 5=60$ — Check by multiplying back: 12 workers for 5 days is the same 60, so the answer is right.

Worked example

1. 1

   $3\times 80=240\text{ km}$ — The distance: 3 hours at 80 km/h. Time × speed gives distance, and the trip is the same length at any speed.

2. 2

   $240÷60=4\text{ hours}$ — Time = distance ÷ speed, so divide by the new speed. Slower than 80 km/h, so it takes longer: 4 hours, not 3.

3. 3

   $240÷2.5=96\text{ km/h}$ — Speed = distance ÷ time, so divide by 2.5 hours. Less time needs a faster speed: 96 is more than 80.

Worked example

1. 1

   $12\times 20=240$ — The hay: 12 horses for 20 days is 240 horse days. Multiply first, because the amount of hay does not change.

2. 2

   $240÷15=16\text{ days}$ — Share the same hay among 15 horses, so divide by 15. More horses, so it lasts fewer days: 16, not 20.

Worked example

1. 1

   $20-5=15\text{ days}$ — After 5 days, the 12 horses still had 15 days of hay left.

2. 2

   $12\times 15=180$ — The hay left: 12 horses for those 15 days is 180 horse days. Only this is shared from now on.

3. 3

   $12+3=15\text{ horses}$ — The number of horses eating from now on.

4. 4

   $180÷15=12\text{ more days}$ — Share only the hay left among 15 horses, so divide by 15.

5. 5

   $5+12=17\text{ days}$ — If the question asks how long the hay lasted altogether, add back the first 5 days.

Worked example

1. 1

   $x_1{y}_1=x_2{y}_2$ — Write the rule first. It shows the examiner your method.

2. 2

   $9\times 8=6\times d$ — Put in the numbers. d is the unknown number of days for 6 workers.

3. 3

   $72=6d$ — 9 × 8 = 72. This is the total job, 72 days of work.

4. 4

   $d=72÷6=12\text{ days}$ — Divide both sides by 6 to get d alone.

5. 5

   $12>8$ — Check the direction: fewer workers, so more days. It makes sense.

Worked example

1. 1

   $5\times 18=90$ — The whole job: 90 days of work.

2. 2

   $90÷8=11.25$ — Divide by the 8 days to find the number of builders.

3. 3

   $11.25\to 12\text{ builders}$ — You cannot have a quarter of a builder, so 11.25 is impossible. 11 builders would take $90÷11\approx 8.2$ days, too long, so round up to 12.

Worked example

1. 1

   $7\times 20=140$ — The food: 140 dog days.

2. 2

   $140÷9=15.6$ — Share it among 9 dogs.

3. 3

   $15.6\to 15\text{ full days}$ — On day 16 the food runs out part way, so only 15 full days. Round down here.

Worked example

1. 1

   $4\times 5=20\text{ machine hours}$ — 4 machines for 5 hours is 20 machine hours of work, and that made 600 bottles.

2. 2

   $600÷20=30$ — Each machine makes 30 bottles in one hour. Divide the bottles by the machine hours.

3. 3

   $1200÷30=40\text{ machine hours}$ — 1200 bottles need 40 machine hours of work.

4. 4

   $40÷5=8\text{ hours}$ — 5 machines share the 40 machine hours, so divide by 5.