Ray's New Higher Arithmetic · Arts. 79–81 · Unit 6: Division
19. Short cuts in division
Goal: Your student divides by a composite number using its factors and finds the true remainder, and divides quickly by 10, 100, 1000 and by numbers ending in ciphers.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Contractions in division: Case I, a composite divisor (217 ÷ 15 by dividing by 3 then 5, with the true remainder); Case II, a divisor of 1 with ciphers (23543 ÷ 100); Case III, ciphers on the right of the divisor (3846 ÷ 400).
The lesson, step by step
Warm-up
Mental: 4500 ÷ 100; 4500 ÷ 9; 4500 ÷ 90.
Composite divisor
Read:
When the divisor is a composite number.
Divide the dividend by one factor of the divisor, and divide this quotient by another factor, and so on.
Work the book's example: 217 ÷ 15. 217 ÷ 3 = 72, remainder 1. 72 ÷ 5 = 14, remainder 2. The true remainder is 2 × 3 + 1 = 7. So 14, remainder 7. Check: 14 × 15 + 7 = 217.
Why the remainder needs fixing
The second remainder, 2, counts threes, not units. So it is worth 2 × 3 = 6 units, plus the first remainder of 1.
Ciphers
Read:
Cut off as many figures in the dividend as there are ciphers in the divisor;
Work: 23543 ÷ 100 = 235, remainder 43. Then the book's Case III example: 3846 ÷ 400. Cut off two figures: 38, remainder 46. 38 ÷ 4 = 9, remainder 2. True remainder 2 × 100 + 46 = 246.
Practice
Your student makes up and solves three of each kind, then checks each by multiplying back. Try 1000 ÷ 24 using 4 × 6, and 98765 ÷ 1000.
Wrap-up
Ask: What is 5000 ÷ 250 in your head? (500 ÷ 25 = 20.)
Answers to the book's problems
Practice: 1000 ÷ 24 = 41, remainder 16 (1000 ÷ 4 = 250; 250 ÷ 6 = 41 r 4; true remainder 4 × 4 + 0 = 16). 98765 ÷ 1000 = 98, remainder 765.
Worked out for this site from the scan. If a number in your copy differs, trust the book.