Ray's New Higher Arithmetic · Arts. 88–90 · Unit 6: Division

22. Short cuts with parts of 100

Goal: Your student multiplies by numbers like 25, 33⅓ and 87½ using fractions of 100, multiplies by numbers with all-alike digits, and divides by numbers ending in a simple part of 100.

⏱ About 40 minutes · 7 steps

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You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil

What's in the book

Contractions in multiplication and division: Case I, multiplying by a simple part of 100 or 1000 (246 × 87½); Case II, multiplying by a number whose digits are all alike (592643 × 66666); Case III, dividing by a number ending in a simple part of 100 (6903141128 ÷ 21875). Examples follow, most with answers.

The lesson, step by step

  1. Warm-up

    Mental: 44 × 25 (think 4400 ÷ 4). 36 × 50.

  2. Parts of 100

    Read:

    Multiply by 100, 1000, etc., and take such a part of the result as the multiplier is of 100, 1000, etc.

    Work: 246 × 87½. 87½ is 7/8 of 100. 24600 × 7/8 = 21525.

  3. Guided practice

    Together:

    6564 X 62 1/2

    62½ is 5/8 of 100: 656400 ÷ 8 × 5 = 410250, as printed.

  4. Digits all alike

    Read:

    To multiply by any number whose digits are all alike.

    Multiply as if the digits were 9 (easy: add ciphers and subtract), then take the right fraction. For 66666, take 2/3 of the product by 99999. Try the first example: 451402 × 3333. (1504522866.)

  5. Dividing

    Read the rule for Case III:

    Multiply both dividend and divisor by such a number as will convert the final figures of the divisor into ciphers

    For a divisor of 225, multiply both by 4: the divisor becomes 900. Then:

    If there be a remainder, it should be divided by the multiplier, to get the true remainder.

  6. Independent practice

    Your student works the examples for practice. Examples 1 to 3 of Case III have no printed answers; they are below.

  7. Wrap-up

    Ask: What part of 100 is 12½? (1/8.) 16⅔? (1/6.)

Answers to the book's problems

Case III examples: 1) 300521761 ÷ 225 = 1335652, remainder 61. 2) 1510337264 ÷ 43750 = 34521, remainder 43514. 3) 22500712361 ÷ 1406250 = 16000, remainder 712361. (Check that the numbers on your page match these; the scan is faint.)

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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