(1.3125)10 is represented as 1.5 in which of the following bases?

Concept

In any base b, a numeral is read by place value: the digit before the point counts whole units, and the first digit after the point counts b-ths of a unit.

So the numeral 1.5 written in base b stands for the decimal quantity 1 + 5/b. A digit is legal only when it is smaller than its own base, so the digit 5 can appear only in a base greater than 5.

Application

1. Call the unknown base b and translate the numeral: 1.5 in base b means 1 + 5/b.

2. The question says this quantity equals 1.3125 in decimal, so write 1 + 5/b = 1.3125.

3. Subtract 1 from both sides: 5/b = 0.3125.

4. Turn the decimal into a fraction: 0.3125 = 3125/10000 = 5/16.

5. Compare 5/b with 5/16. The numerators are already equal, so the denominators must be equal: b = 16.

6. Confirm the digit is legal: 5 is smaller than 16, so 5 is a valid digit in this base.

Cross-check

Convert in the other direction, by repeatedly multiplying the fractional part by the base: 0.3125 x 16 = 5.0. The whole part 5 becomes the first digit after the point and nothing is left over.

(1.3125)10 = (1.5)16.

Testing every offered base the same way:

Only base 16 reproduces 1.3125, so the required base is 16, that is hexadecimal.