Question

In a box, there is a total of 82 fifty-cent coins and twenty-cent coins. The total value of the twenty-cent coins is \$1.00 more than the total value of the fifty-cent coins. How many fifty-cent coins are there?

Source: Anderson Primary

Assume 82 20c, 0 50c

Base case

82 x 20c = 1640             }

0 x 50c    = 0                   } difference = 1640

Next case 81 20c, 1 50c

81 x 20c = 1620             }

1 x 50c    =     50             } difference = 1570

Finding pattern

1640 – 1570 = 70          (every time decreases by 70)

Target case:

We need to have difference of 100 or \$1 as stated in the question.

We need to know 1640 – 70 how many times to get to 100

1640 – 100 = 1540       (base case of 1640 – target case of 100)

1540 ÷ 70 = 22               (22 times)                                                                                               Ans: 22

More explanation and cross checking:

22 times would mean 0 x 50c now becomes 22 x 50c

82 – 22 = 60                    (22 50c means 60 20c)

60 x 20c = 1200             (total value of 20c)

22 x 50 c = 1100            (total value of 50c)

1200 – 1100 = 100       (difference is \$1, confirm correct 😊)

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1/0.20 = 5
82 – 5 = 77
2 + 5 = 7 (2 50c coins = 5 20c coins)
77/7 = 11
11 x 2 = 22

Ans : 22 fifty-cent coins.

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Total we have 82 coins (50¢ & 20¢).

2 50¢ = 5 20¢.
In order for the number of 50¢ and 20¢ to be equal, there should be sets of 7 coins (2 50¢, 5 20¢)
82/7 = 11R5
This implies there are 11 sets of 2 50¢ coins and 5 20¢ coins with 5 coins leftover.

For total value of 20¢ to be \$1 more than 50¢, there needs to be 5 more 20¢ than 50¢.
Total number of 20¢ coins = 11 x 5 + 5 = 60
Total number of 50¢ coins = 11 x 2 = 22

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Can use guess-and-check method:

30x 50c coins     Value = \$15.00        52x 20c coins         Value = \$10.40

20x 50c coins     Value = \$10.00        62x 20c coins         Value = \$12.40

22x 50c coins     Value = \$11.00        60x 20c coins          Value = \$12.00

Number of 50-cent coins = 22 (ans)

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