Olympiad topic · P3–P6
Sum and multiple
Sum-and-multiple and difference-and-multiple questions appear in the Hua Xia Cup's P3 to P5 first rounds, qualifying rounds and finals.
Try one
★★ MediumAt Chinese New Year two sisters got $218 in red-packet money altogether.
The older sister got $38 more than twice what the younger got.
How much did the older sister get?
- Hint 1: Without her extra $38, the older would have exactly twice as much. What would the total be then?
- Hint 2: Then the younger is 1 part and the older 2 parts.
- 1.Call the younger's money 1 part: the older has 2 parts plus $38. Together they have $218.
- 2.Take away the older's extra $38 and she has exactly 2 parts: 218 − 38 = 180.
- 3.That is 3 parts, so the younger has 180 ÷ 3 = $60.
- 4.The older has 218 − 60 = $158.
- ⚠️Common mistake: adding the $38. The older has $38 extra, so take it away first.
Answer: 158 dollars
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsTotal-and-times or difference-and-times in one step.
- ★★MediumFree2 question types · unlimited questions"34 more than twice", to adjust first.
- ★★★HardFree2 question types · unlimited questionsThree numbers linked in a chain, or the multiple changing after some are taken.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsTwo "what if" cases compared, or place value mixed in.
Sum and multiple: what it is and how to solve it
You know two numbers' total (or difference) and that one is so many times the other; find each.
For example: A tank has 24 goldfish and tropical fish; there are 3 times as many goldfish. How many tropical fish? 24 ÷ (3 + 1) = 6.
The key idea: Call the smaller number 1 part; the larger is that many parts: total ÷ (times + 1) = 1 part, or with a difference, difference ÷ (times − 1). The usual method at school uses an equation: let there be x tropical fish, x + 3x = 24.
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