Decimal representation

The governing concept is converting equivalent forms before performing subtraction. Convert the fraction to a decimal by dividing 13 by 32. Since 1/32 = 0.03125, multiplying by 13 gives 13/32 = 0.40625. Now subtract the given decimal: 0.40625 − 0.15625 = 0.25000, which is 0.25. Therefore option A is correct. A useful check is to write both numbers with five decimal places; the subtraction is 40625 − 15625 in units of 1/100000, giving 25000/100000 = 0.25. Option B repeats the subtracted number, option C is only the converted fraction before subtraction, and option D is unrelated to the required difference.

The governing concept is multiplication of a terminating decimal by a power of 10. Since 1,000,000 = 10⁶, the decimal point in 0.0078125 moves six places to the right. Counting the shifts gives 0.078125 after one, 0.78125 after two, 7.8125 after three, 78.125 after four, 781.25 after five, and 7812.5 after six shifts. Therefore 0.0078125 × 1,000,000 = 7812.5, so option C is correct. Option A corresponds to moving the point only four places, option B to five places, and option D to seven places or an extra shift. The power-of-ten rule and the explicit place-value count independently confirm the answer.

The governing concept is comparison of decimal numbers by place value. Write the numbers as 0.6a4 and 0.674. Their tenths digits are both 6, so the comparison moves to the hundredths place. In 0.6a4, that digit is a; in 0.674, it is 7. To make the first number greater, we need a > 7. Since a is a digit, the possible values are 8 or 9, and the smallest is 8. Therefore option C is correct. If a = 6, then 0.664 is smaller than 0.674; if a = 7, the numbers become 0.674 and 0.674, which are equal rather than greater. Although 9 also works, it is not the smallest possible value.

The governing concept is solving a linear equation by applying the inverse operation. Since 10000x means 10000 multiplied by x, divide both sides by 10000: x = 1.1875 ÷ 10000. Dividing by 10,000 moves the decimal point four places to the left: 1.1875 → 0.11875 → 0.011875 → 0.0011875 → 0.00011875. Hence option C is correct. A quick check confirms it: 0.00011875 × 10000 = 1.1875. Option A results from shifting in the wrong direction, option B has only two decimal-place shifts, and option D has only one shift. The equation therefore requires the value shown in option C.

The governing concept is preserving a quotient while clearing decimal points. Multiply both dividend and divisor by 1000, which does not change their ratio: 0.567 ÷ 0.021 = 567 ÷ 21. Now divide 567 by 21. Since 21 × 27 = 567, the quotient is 27. Therefore option B is correct. A direct check gives 0.021 × 27 = 0.567, confirming the result. The answer 2.7 is ten times too small, 270 is ten times too large, and 0.27 is smaller by a factor of 100. The important point is that both numbers must be multiplied by the same power of 10; moving the decimal in only one number would change the value of the quotient.

The governing concept is place-value movement when multiplying a decimal by a power of 10. Since 1,000,000 = 10⁶, multiplication moves the decimal point six places to the right. Starting with 0.00390625, the shifts are 0.0390625, 0.390625, 3.90625, 39.0625, 390.625, and finally 3906.25. Therefore option C is correct. The same result can be checked by writing 0.00390625 = 390625/100000000 and multiplying by 1000000, which gives 3906.25. Option A corresponds to moving the point four places, option B to five places, and option D to seven places. Counting all six zeros prevents these place-value errors.

The governing concept is recurring-decimal notation. A bar is placed only above the complete block of digits that repeats indefinitely; non-repeating digits before that block remain outside the bar. In 0.00252525..., the first two digits after the decimal point are 00, and the repeating part is 25: 0.00 25 25 25 ... . Consequently, the correct notation is 0.00 overline{25}, which is option B. Option A incorrectly includes the initial non-repeating zeros in the recurring block. Option C marks only 5, although the repeating cycle has two digits, 25. Option D treats 025 as the repeating block, which does not match the displayed decimal expansion. Therefore B is the only unambiguous answer.

The governing concept is decimal division: multiplying both dividend and divisor by the same power of 10 does not change the quotient. Since 0.013 has three digits after the decimal point, multiply both numbers by 1000. Thus 0.819 ÷ 0.013 = 819 ÷ 13. Because 13 × 63 = 819, the quotient is 63, so option B is correct. The decimal shift must be applied equally to both numbers; shifting only one number would alter the problem. Option A and option D are ten times smaller than the correct result because of an incorrect shift or placement of the decimal point. Option C is ten times larger and commonly comes from shifting in the opposite direction too far.

The governing rule is that a rational number has a decimal expansion that terminates or eventually repeats a fixed block of digits. In option C, the block 45 repeats forever, so 0.\overline{45} = 45/99 = 5/11, which is a ratio of integers and therefore rational. The other displayed decimals are described as non-terminating and non-repeating. Such decimal expansions cannot be written as p/q and are irrational. The important test is not merely whether the decimal continues indefinitely: some endless decimals are rational when they repeat, while non-repeating endless decimals are irrational. Therefore option C is the only valid answer.

A non-terminating, non-recurring decimal continues without end and never settles into a fixed repeating cycle. Option B, 0.4040040004…, has an increasing and changing number of zeros between successive 4s, so no fixed block repeats. It is therefore non-terminating and non-recurring, representing an irrational number. Option A repeats the block 27 and is rational. Option C terminates and equals 1/8. Option D repeats 8 and equals 8/9, so it is also rational. The correct test is to inspect both features—whether the expansion ends and whether its digits eventually repeat. Only option B satisfies both required conditions.