Olympiad topic · P1–P6
Clever sums
The first few questions of a competition paper are often clever sums, and the Hua Xia Cup has them in every year from P1 to P6.
Try one
★★ MediumWork this out the clever way.
14 × 31 + 31 × 86 = ?
What is the answer?
- Hint 1: Do the two parts share a number?
- Hint 2: Take the shared 31 outside and work out the sum in brackets first.
- 1.Both parts have the factor 31: 31 × (14 + 86).
- 2.The brackets make exactly 100: 31 × 100 = 3100.
Answer: 3100
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsRound numbers, reordering.
- ★★MediumFree2 question types · unlimited questionsConsecutive numbers, the distributive law.
- ★★★HardFree2 question types · unlimited questionsEven runs of numbers, several common factors.
- ★★★★Expert (competition level)VIP5 question types · unlimited questionsSums of products of neighbours, fraction products that cancel a few places apart, sums of squares of an even run, perfect squares hidden in decimals, decimals with a common factor.
Clever sums: what it is and how to solve it
Using the laws of arithmetic and patterns to make long sums easy.
For example: 25 + 49 + 51 + 75 = ? Pair the ones that make round numbers: (25 + 75) + (49 + 51) = 100 + 100 = 200.
The key idea: Make tens and hundreds when adding; look for 2 × 5 and 4 × 25 when multiplying; take out a common factor, e.g. 149 × 98 − 49 × 98 = 100 × 98; an even run of numbers adds to (first + last) × how many ÷ 2.
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