- A8
- B9
- C7
- D63
Step-by-Step Solution
and . The numerators strictly between 18 and 27 are 19, 20, …, 26, which is numbers. The trap is computing , which quietly counts one of the endpoints.
Answer: A) 8
How many rational numbers with denominator 63 lie strictly between and ?
Class 8Rational NumbersRational Numbers
Key idea: Finding rationals between two rationals
**Easiest way: the average.** Between a and b, the rational (a + b) / 2 lies exactly in the middle. Repeat to find more: between a and (a+b)/2 lies their average, and so on — forever.
**Common-denominator way**: bring both rationals to a common denominator with extra room. Between 1/3 and 1/2, write as 2/6 and 3/6 — but no integer fits between 2 and 3. Multiply top and bottom of both by 10: 20/60 and 30/60. Now 21/60, 22/60, …, 29/60 all sit between — nine fresh rationals.
This density is something integers don't have: there's no integer between 4 and 5.
**Common-denominator way**: bring both rationals to a common denominator with extra room. Between 1/3 and 1/2, write as 2/6 and 3/6 — but no integer fits between 2 and 3. Multiply top and bottom of both by 10: 20/60 and 30/60. Now 21/60, 22/60, …, 29/60 all sit between — nine fresh rationals.
This density is something integers don't have: there's no integer between 4 and 5.
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