Unitary Method

Problem. If 8 litres of milk cost Rs 720, what do 15 litres cost?

Solution. Cost of 1 litre =720\div8=\text{Rs }90.

Cost of 15 litres =15\times90=\text{Rs }1350.

Answer. Rs 1,350.

Check. 720:1350=8:15 ✓ — the ratio of the quantities equals the ratio of the costs, which is what direct variation means.

Problem. A car covers 156 km in 3 hours at a steady speed. How far will it go in 7 hours, and how long will it take to cover 260 km?

Solution. Distance in 1 hour =156\div3=52 km.

In 7 hours: 7\times52=364 km.

For 260 km: 260\div52=5 hours.

Answer. 364 km; 5 hours.

Check. 52\times5=260 ✓, and the speed 52 km/h is the same throughout.

Problem. 12 men can complete a piece of work in 10 days. How many days will 15 men take to do the same work?

Solution. Total work =12\times10=120 man-days.

With 15 men, the time is 120\div15=8 days.

Answer. 8 days.

Check. 12\times10=15\times8=120 ✓. The answer is smaller than 10, as it must be when more men are put on the same work.

Problem. 8 men working 6 hours a day finish a job in 15 days. How many days will 10 men take, working 9 hours a day?

Solution. Total work in man-hours:

With 10 men at 9 hours a day, the daily work is 10\times9=90 man-hours.

Answer. 8 days.

Check. 10\times9\times8=720 ✓ — the same total work as 8\times6\times15.

Problem. Two taps fill a tank in 12 hours and 18 hours respectively. How long will they take to fill it together?

Solution. Work in "tank per hour". The first tap fills \frac{1}{12} of the tank in an hour, the second \frac{1}{18}.

Together in one hour:

of the tank. So the whole tank takes

Answer. 7 hours 12 minutes.

Check. In 7.2 hours the first tap supplies \frac{7.2}{12}=0.6 of the tank and the second \frac{7.2}{18}=0.4, and 0.6+0.4=1 ✓. The answer must lie between 6 hours (if both were as fast as the first) and 12 hours.

Problem. A barrack has food for 200 soldiers for 30 days. How many soldiers must be transferred so that the food lasts 40 days?

Solution. Fewer soldiers → food lasts longer: inverse variation. Food =200\times30=6000 soldier-days. Soldiers who can be fed for 40 days =\frac{6000}{40}=150. Soldiers to transfer =200-150= 50.

Problem. 20 workers can finish a job in 15 days. How many more workers must be added to finish it in 12 days?

Solution. Work =20\times15=300 worker-days. Workers needed for 12 days =\frac{300}{12}=25. Workers to add =25-20= 5.

Problem. 10 people earn Rs 1,60,000 in 20 days. How much will 15 people earn in 10 days at the same rate?

Solution.

More people → more income (D). More days → more income (D).

Unitary check. One person in one day earns \frac{1\,60\,000}{10\times20}= Rs 800; 15 people in 10 days earn 800\times15\times10= Rs 1,20,000 ✓.

Problem. 30 workers build a 600 m wall in 20 days. How long a wall will 15 workers build in 18 days?

Solution. Workers and length: D. Days and length: D.

Problem. 10 people build a bamboo shelter in 14 days working 8 hours a day. How many hours a day must 7 people work to build the same shelter in 16 days?

Solution. Fewer people → more hours per day (I). More days → fewer hours per day (I).

Check. Total work =10\times14\times8=1120 person-hours, and 7\times16\times10=1120 ✓.

Problem. A bus travels at 54 km/h. Express this in m/s and find how far it goes in 25 seconds.

Solution. 54\text{ km/h}=54\times\frac{1000}{3600}=54\times\frac{5}{18} =15 m/s.

Distance in 25 s =15\times25=375 m.

Answer. 15 m/s; 375 m.

Check. 375 m in 25 s is 15 m/s, and 15\times3.6=54 km/h ✓.

Problem. A man walks 4 km at 5 km/h and then travels 12 km by bus at 40 km/h. Find his total time and his average speed for the whole journey.

Solution. Walking time =\frac{4}{5}=0.8 h.

Bus time =\frac{12}{40}=0.3 h.

Total time =1.1 h =1 hour 6 minutes.

Total distance =16 km, so

Answer. 1.1 hours; about 14.55 km/h.

Check. The average speed must lie between 5 and 40 km/h and must be much closer to 5, because the walking took nearly three times as long as the bus ride. It is 14.55, well below the midpoint of 22.5 ✓.

Question. 12 workers build 150 m of wall in 10 days. How many days will 15 workers take to build 225 m of the same wall?

Answer. Work with one worker and one day at a time. In one day 12 workers build \frac{150}{10} = 15 m, so one worker builds \frac{15}{12} = 1.25 m a day. 15 workers build 15 \times 1.25 = 18.75 m a day. Days needed = \frac{225}{18.75} = \textbf{12 days}. Check: more workers but much more wall, so more days than 10 — the answer is believable.