Asked by nooo
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Part 1
Reasoning A circuit board is one fifth
inch wide. How many StartFraction 1 Over 20 EndFraction
-inch-wide circuits fit on the board? Use a number line to find the solution. Use pencil and paper. Is the quotient of two unit fractions always greater than the divisor? The dividend? Explain.
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Part 1
The board can fit
enter your response here circuits on it.
(Simplify your answer.)
All Answers 1
Answered by
GPT-5 mini
AI
The board can fit 4 circuits on it.
Reason: (1/5) ÷ (1/20) = (1/5)·(20/1) = 4. On a number line from 0 to 1/5 mark off 1/20 increments: 1/20, 2/20, 3/20, 4/20 = 1/5 — four pieces.
About unit fractions in general: (1/a) ÷ (1/b) = b/a.
- Is the quotient always greater than the divisor (1/b)? No. b/a > 1/b iff b^2 > a, which is not always true (counterexample: (1/10) ÷ (1/2) = 1/5 < 1/2).
- Is the quotient always greater than the dividend (1/a)? Yes except when b = 1. Since b/a > 1/a exactly when b > 1, the quotient is ≥ the dividend (equal only when b = 1).
Reason: (1/5) ÷ (1/20) = (1/5)·(20/1) = 4. On a number line from 0 to 1/5 mark off 1/20 increments: 1/20, 2/20, 3/20, 4/20 = 1/5 — four pieces.
About unit fractions in general: (1/a) ÷ (1/b) = b/a.
- Is the quotient always greater than the divisor (1/b)? No. b/a > 1/b iff b^2 > a, which is not always true (counterexample: (1/10) ÷ (1/2) = 1/5 < 1/2).
- Is the quotient always greater than the dividend (1/a)? Yes except when b = 1. Since b/a > 1/a exactly when b > 1, the quotient is ≥ the dividend (equal only when b = 1).
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