A hard question on fraction problems for P5–P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 qualifying rounds and finals; harder ones work backwards through fractions, or need clever shortcuts with fractions.
The question
Wing Sze borrowed a novel.
On the first day Wing Sze read 5 pages fewer than a sixth of the book.
On the second day Wing Sze read 7 pages fewer than four fifths of what was left.
That left 38 pages unread.
How many pages does the novel have?
Show the hints (3)
💡Hint 1: Work backwards: how many pages were unread before the second day?
💡Hint 2: Split the pages left at the start of the second day into five equal parts: that day took four parts minus 7 pages, so the 38 left are the other parts plus 7 pages. Find one part first.
💡Hint 3: Once you have the pages at the start of the second day, do the same again for the first.
Worked solution
Have a go first, then read the solution step by step.
1.Work backwards, starting with day 2: split the pages left at the start of day 2 into 5 equal parts. Wing Sze read 4 parts minus 7 pages, so the 38 pages left are 1 part plus 7 pages.
2.The 7 pages not read are inside the 38 left, so take them away: 1 part = 38 − 7 = 31 pages.
3.One part is 31 pages. At the start of day 2 there were 31 × 5 = 155 pages.
4.Now day 1, the same way: split the whole book into 6 equal parts. Wing Sze read 1 part minus 5 pages, so the 155 pages left are 5 parts plus 5 pages.
5.The 5 pages not read are inside the 155 left, so take them away: 5 parts = 155 − 5 = 150 pages.
6.One part = 150 ÷ 5 = 30 pages. The whole book has 30 × 6 = 180 pages.
7.Check: 180 pages → 25 read on day 1, 155 left → 117 read on day 2, 38 left ✓
⚠️Common mistake: taking day 2's 4/5 as 4/5 of the whole book, or, working back, forgetting to add back pages read "more" and take away pages read "fewer".
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download