Time and work, speed and distance, and interest

Time and work, speed and distance, and interest questions all depend on a rate: work per day, distance per hour or interest per year. Turn every question into a rate first, and the arithmetic becomes simple.

Time and work

If a person finishes a job in n days, they do 1/n of it each day. Add the rates when people work together, then take the reciprocal.

A: 12 days -> 1/12 per day
B:  6 days -> 2/12 per day
Together   -> 3/12 = 1/4 per day -> 4 days

Pipes work the same way, but an emptying pipe is subtracted. A fills in 6 hours and B empties in 9 hours: 1/6 - 1/9 = 1/18, so the tank fills in 18 hours.

A quick check: the combined time must be less than the faster person's time when both are working.

Speed, distance and time

• Speed = distance / time.
• km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.
• A train passing a pole covers its own length. Passing a platform, it covers its length plus the platform.

Average speed is total distance divided by total time, not the average of the speeds. Going 60 km at 30 km/h and returning at 60 km/h takes 2 + 1 = 3 hours for 120 km, so the average is 40 km/h, not 45.

Simple and compound interest

Simple interest = P x R x T / 100. Interest is the same every year because it is always calculated on the original principal.

Compound interest adds each year's interest to the principal. Amount = P x (1 + R/100)^T, and interest = amount - P.

P = 10,000, R = 10%, T = 2
Year 1: 10,000 + 1,000 = 11,000
Year 2: 11,000 + 1,100 = 12,100
CI = 2,100   (SI would be 2,000)

Averages

Work with totals. Total = average x count. If a number is removed or added, change the total first and then divide by the new count.

Checklist: write the rate, check the units match, calculate, then test the answer against common sense. Two workers together must be faster than either alone, and compound interest must be higher than simple interest after the first year.