Time and Work Questions in JNVST Arithmetic, Solved Without Memorized Shortcuts

The Shortcut That Works Until It Doesn't

Time and work questions are one of the few places in JNVST arithmetic where a memorised shortcut, "multiply the two times and divide by their sum," gets taught before the underlying idea, and that ordering causes real damage the moment a question deviates even slightly from the two-person, start-together version of the problem. Going through past papers, the questions that trip students up are the ones with three workers, or with someone leaving partway through, where a plugged-in shortcut simply has no slot for the extra condition.

A Worked Example With Combined Work

Take this: "A can finish a piece of work in 10 days, and B can finish the same work in 15 days. They work together for 3 days, and then A leaves. In how many more days will B finish the remaining work?" A student relying purely on the combined-time shortcut has no way to handle the "A leaves after 3 days" condition, because that shortcut only answers "how long together," not "how much is left after some days together." The correct approach treats the whole work as 1 unit, gives A a rate of one-tenth per day and B a rate of one-fifteenth per day, adds those to get a combined rate of one-sixth per day, and finds that in 3 days together they complete 3 times one-sixth, which is half the work. The remaining half is left for B alone, and at B's rate of one-fifteenth per day, finishing half the work takes 7.5 days.

The Trap: Adding Days Instead of Adding Rates

The most common error in this chapter is trying to combine the "10 days" and "15 days" by adding or averaging the day counts directly, rather than converting each to a rate of work per day first. Days cannot be added to represent combined work, because more workers reduce the number of days needed, they do not add on top of each other; only the rates, work per day, add together correctly.

Why Adding Days Feels Like the Obvious Move

Combining two given numbers by adding them is the most natural first instinct in arithmetic, and 10 and 15 are the two numbers the question hands over directly, so adding or averaging them feels like using the given information. The actual rule, that work rates add while completion times do not, is counterintuitive enough that it needs to be applied deliberately rather than assumed.

A Habit Built Around "Work Done in One Day"

For every time and work question, before touching the specific numbers in the question, have your child write "work done by each person in 1 day" as a fraction, for every person mentioned:

1. Convert each person's total days into a one-day work fraction.
2. Add the relevant fractions for however many people are working together at each stage of the question.
3. Only then bring in the specific number of days or amount of work the question actually asks about.

Starting from the one-day fraction every time removes the temptation to shortcut straight to a memorised formula that only fits the simplest version of the question.