Decimal representation

( \frac{37}{128}=0.2890625 ), so (0.2890625-0.1640625=0.125). Convert the fraction into decimal and subtract.

The governing concept is that multiplying both dividend and divisor by the same nonzero power of 10 does not change their quotient. To remove the decimal point from the divisor, multiply both numbers by 1000: 0.945×1000=945 and 0.015×1000=15. Thus 0.945÷0.015=945÷15. Since 15×63=945, the quotient is 63, so option B is correct. A place-value check gives the same result: 0.015×60=0.900 and 0.015×3=0.045, whose sum is 0.945. Option A is ten times too small, option C is ten times too large, and option D results from shifting the decimal point incorrectly. The essential rule is to scale both numbers equally, not just the divisor.

The first digit after the decimal point, 0, occurs only once. After it, the block 96 repeats as 96, 96, 96, … . Therefore, the bar is placed only over 96: \(0.0\overline{96}\). In \(0.\overline{96}\), 0 would also repeat, while \(0.09\overline{6}\) makes only 6 recurring. Exam tip: write out a few decimal digits and identify the shortest repeating block before placing the bar.

\(\frac{21}{30}=\frac{7}{10}=0.7\), so its decimal expansion terminates. The prime-factor test is applied only after reducing the fraction; in \(\frac{7}{30}\), factor 3 remains. Exam tip: simplify first.

To make (2^{10}) into (10^{10}), we multiply by (5^{10}). So there will be at most (10) decimal places.

The denominator \\(2^5\times5^8\\) contains only the prime factors 2 and 5, so the fraction can have a terminating decimal after simplification. To convert the denominator into a power of 10, the five factors of 2 are paired with five factors of 5. Three factors of 5 remain, and supplying three factors of 2 gives \\(2^8\times5^8=10^8\\).

Therefore the greatest possible number of decimal places is eight, which is option B. The rule is to use the larger exponent, \\(\max(5,8)=8\\), rather than adding the exponents to get 13. Cancellation involving the numerator can make the decimal shorter, but it cannot require more than eight places under the stated simplified denominator.

The number of zeros between (8)'s increases, so there is no fixed repetition. Such a decimal is non-terminating non-recurring.

To convert a decimal into a percentage, multiply it by 100: \(0.0048 \times 100 = 0.48\). Therefore, the correct answer is 0.48%. The closest distractor, 4.8%, would result from converting 0.048 into a percentage, not 0.0048. Exam tip: Move the decimal point two places to the right when converting a decimal to a percentage.

On multiplying by (10000000), the decimal point moves seven places to the right. Therefore, the answer is (9765.625).

The bar over 9 means that 9 repeats forever: 0.249̅ means 0.2499999… rather than 0.249 only. A fundamental decimal identity is 0.2499999… = 0.2500000…, because an infinite tail of 9s carries into the preceding digit. Also, 1/4 = 0.25 exactly. Therefore all three quantities have the same value, so option C is correct. Option A incorrectly treats the repeating decimal as a finite 0.249, while options B and D incorrectly place one of the equal numbers strictly between the others. The equality can also be checked by writing 0.25 − 0.249999… = 0.

( \frac{19}{32}=0.59375 ), so (0.59325) is greater than (0.593) and less than (0.59375). Convert boundary values into decimals.

In lowest terms, a denominator containing only 2 and 5 can be converted into a factor of \(10^n\), so the decimal terminates. Any other prime factor gives a recurring decimal. Exam tip: reduce the fraction first before checking its denominator.

(64000\times15625=1000000000), so ( \frac{257}{64000}=0.004015625 ). Convert the denominator into a power of (10) to get the exact decimal.

(0.000234375=\frac{234375}{1000000000}=\frac{3}{12800}). Count decimal places and write the fraction in simplest form.

The tenths digits are equal and the hundredths digit must satisfy (a<3). So (a=0,1,2), giving (3) possible values.

The governing concept is recurring-decimal notation: a bar is placed over exactly the digit or block that repeats indefinitely. In 0.000727272…, the first three digits after the decimal point are 000 and do not repeat as part of the recurring cycle. After them, the two-digit block 72 repeats: 0.000 72 72 72… . Therefore the correct notation is 0.000̅72, with the bar covering both 7 and 2, so option B is correct. Option A places the bar over non-repeating digits and does not represent the actual cycle. Option C shifts the recurring block and includes an extra zero. Option D places the bar only over 2, although 7 also repeats in every cycle. Correct bar placement must identify the complete minimal repeating block.

(5120\times1953125=10000000000), so ( \frac{191}{5120}=0.0373046875 ). Converting a large denominator into a power of (10) is an exact method.

(0.00025390625=\frac{25390625}{100000000000}=\frac{13}{51200}). Count decimal places and write the fraction in simplest form.

The denominator has (23), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

The governing concept is the denominator test for the decimal expansion of a rational number. First reduce the fraction: the greatest common divisor of 264 and 440 is 88, so 264/440=3/5. In lowest terms, a rational number has a terminating decimal exactly when the denominator contains no prime factors other than 2 and 5. The reduced denominator is 5, so the condition is satisfied. Indeed, 3/5=0.6, which terminates after one decimal place. Therefore option B is correct. Option A would be appropriate if another prime factor, such as 3 or 7, remained in the denominator. Option C describes a non-terminating non-recurring decimal, while option D is false because 0.6 is not an integer. Simplification must be done before applying the test.