18. Word problem translation
Remainder chains, expected vs actual comparisons, counting equations in DS setups.
Core ideas
- Translate every sentence into an equation before computing anything; name your variables explicitly.
- Remainder chains: n leaves remainder r when divided by d means n = d*k + r. List candidates from the larger divisor and test against the other condition.
- Expected vs actual setups (fewer people, each pays more) become one equation: total/n plus the change equals total/(new n).
- With answer choices available, plugging in candidates is often faster than solving algebraically.
Worked example 1
A positive integer n leaves remainder 3 when divided by 7 and remainder 5 when divided by 8. What is the smallest such n?
Show solution
Write n = 7k + 3, giving candidates 3, 10, 17, 24, 31, 38, 45, ... Test each against the second condition, remainder 5 when divided by 8: 3 no, 10 gives 2, 17 gives 1, 24 gives 0, 31 gives 7, 38 gives 6, 45 gives 5, yes. So n = 45. From here, all solutions are 45 plus multiples of 56 (the LCM of 7 and 8).
Worked example 2
A group planned to split a $180 bill equally, but 3 people dropped out, and each remaining person paid $3 more than planned. How many people were in the original group?
Show solution
Let n be the original size. Planned share = 180/n; actual share = 180/(n - 3); the actual is $3 more: 180/n + 3 = 180/(n - 3). Multiply through by n(n - 3): 180(n - 3) + 3n(n - 3) = 180n, so 3n^2 - 9n - 540 = 0, or n^2 - 3n - 180 = 0, which factors as (n - 15)(n + 12) = 0. So n = 15. Check: planned 180/15 = 12, actual 180/12 = 15, which is $3 more.
Practice set
Question 1
In 5 years, Tom will be twice as old as Sara was 3 years ago. If Tom is now 21 years old, how old is Sara now?
Attach the time shifts to the right person before setting the equation.
Tom in 5 years is 26. Sara 3 years ago was S - 3, where S is her age now. So 26 = 2(S - 3), giving 26 = 2S - 6, so S = 16. Answer 13 comes from forgetting the 3-year shift, and 10 from mixing up which age is doubled.
Question 2
A contractor planned to complete a job in 20 days, assuming steady progress. After 5 days of work, only 20 percent of the job was done. If the crew continues at this actual pace, the job will be completed how many days later than originally planned?
Find the actual daily rate from the first 5 days, then see how long the remaining work takes.
Actual pace: 20 percent in 5 days, or 4 percent per day. The remaining 80 percent takes 80/4 = 20 more days, for a total of 25 days. That is 25 - 20 = 5 days later than planned. Answer 10 compares the wrong baselines, treating the whole job as if restarted.
Question 3
When the positive integer n is divided by 12, the remainder is 5. What is the remainder when 3n is divided by 12?
Multiplying n by 3 also multiplies its remainder, but the result may exceed 12.
Write n = 12k + 5, so 3n = 36k + 15 = 12(3k + 1) + 3. The remainder is 3. Answer 15 forgets to reduce past 12, and answer 5 assumes the remainder is unchanged by multiplication.
Question 4
When 5 is added to twice a certain number, the result is 19. What is three times the number?
Translate the sentence into an equation first, then answer the question that was actually asked.
Let the number be n. Then 2n + 5 = 19, so 2n = 14 and n = 7. The question asks for 3n = 21. Answer 7 is the number itself, the classic answered-the-wrong-question trap.