Teach: Using Equations ② Problems Involving Quantities and Prices

Using Equations ② Problems Involving Quantities and Prices

The most common equation word problems involve quantities and prices. Form equations by using the relationships “unit price × quantity = price,” “total quantity,” and “total price.”

Crane-and-Turtle Type Problems

In problems of the type “You buy a total of ○ items of two kinds for △ yen,” let the quantity of one kind be x and express the other as “total − x.”

Example ①
You buy a total of 15 snacks, some at 70 yen each and some at 50 yen each. The total cost is 870 yen. How many 70-yen snacks did you buy?
Solution:
① Let the number of 70-yen snacks be x and the number of 50-yen snacks be (15 − x).
② Total cost: 70x + 50(15 − x) = 870
③ Solve: 70x + 750 − 50x = 870 → 20x = 120 → x = 6
　You bought 6 70-yen snacks.

Problems Involving Excesses and Shortages

In the pattern “When ○ items are given to each person, △ remain / △ are lacking,” express the total number of items in two ways and form an equality.

Example ②
Candies are to be distributed to children. If 6 candies are given to each child, 4 remain; if 7 candies are given to each child, there are 3 too few. Find the number of children and the number of candies.
Solution:
① Let the number of children be x.
② Total number of candies:
　“If 6 are given to each child, 4 remain” → 6x + 4
　“If 7 are given to each child, 3 are lacking” → 7x − 3
③ Both expressions represent the same total number of candies, so: 6x + 4 = 7x − 3
④ Solve: −x = −7 → x = 7
⑤ Number of candies: 6×7 + 4 = 46 (candies)
　There are 7 children and 46 candies.

Prices and Money on Hand

The pattern “You have ○ yen, buy △ items at ×× yen each, and receive □ yen in change” also appears frequently. Use the relationship “money on hand = price + change.”

Example ③
A person goes shopping with 1000 yen. After buying x bottles of juice at 80 yen each, the change is 200 yen. Find x.
Solution:
Equality: 1000 ＝ 80x + 200
　80x = 800 → x = 10 (bottles)
Key Points
  • Crane-and-turtle type: let one quantity be x and the other be “total − x”.
  • Excesses and shortages: express the total number of items in two ways and form an equality.
  • Money on hand = price + change.
  • Check whether the answer is an integer and a positive number.

Practice Problems

  1. You buy a total of 8 rice balls, some at 100 yen each and some at 150 yen each. The total cost is 950 yen. How many 100-yen rice balls did you buy?
  2. Pencils are to be distributed to children. If 5 pencils are given to each child, 2 remain; if 6 are given to each child, there are 3 too few. Find the number of children.
  3. You go shopping with 2000 yen and buy x items at 240 yen each. The change is 80 yen. Find x.

Answers and Explanations

  1. Answer: 5
    Explanation: 100x + 150(8−x) = 950 → 100x + 1200 − 150x = 950 → −50x = −250 → x = 5.
  2. Answer: 5 children
    Explanation: 5x + 2 = 6x − 3 → −x = −5 → x = 5.
  3. Answer: x = 8
    Explanation: 2000 = 240x + 80 → 240x = 1920 → x = 8.

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