Decimal representation

(16000\times625=10000000), so ( \frac{149}{16000}=0.0093125 ). Convert the denominator into a power of (10) to get the exact decimal.

(0.00021875=\frac{21875}{100000000}=\frac{7}{32000}). Count decimal places and write the final answer in simplest form.

The bar is only over (81), so after (04), (81) repeats again and again. In bar notation, repeat only the barred part.

(3200\times3125=10000000), so ( \frac{127}{3200}=0.0396875 ). Even a large denominator can be converted into a power of (10).

The direct answer is option B, 3/32000. A terminating decimal can first be written over a power of 10. There are eight digits after the decimal point in 0.00009375, so it equals 9375/100000000. Divide numerator and denominator by 3125: 9375 ÷ 3125 = 3 and 100000000 ÷ 3125 = 32000. Therefore the simplified fraction is 3/32000. Option A, 3/3200, is ten times too large. Option B is correct because it has the exact value in lowest terms. Option C, 9375/1000000, uses only six decimal places and is not equal to the given decimal. Option D, 1/9375, is a reciprocal-like expression and has a completely different value. Exam cue: count every digit after the decimal point before writing the denominator.

The denominator has (19), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

The decimal digits are 5, 6, 0, 9, 0, 9, 0, 9, ... . Here, 56 is the non-repeating part, and the block 09 repeats thereafter. Hence the correct notation is \(0.56\overline{09}\). In \(0.560\overline{9}\), only 9 is treated as recurring, but both 0 and 9 repeat as a block. Exam tip: Write out a few decimal digits first and identify the smallest repeating block before placing the bar.

The direct answer is option A, 4.108272727… . The digits under the bar are 27, so only the block 27 repeats indefinitely. The digits before the bar, 108, occur once. Thus the decimal expands as 4.10827272727… . Option A is correct because it preserves the one-time prefix 108 and then repeats 27. Option B, 4.10827108…, changes the repeating pattern and incorrectly brings back 108. Option C, 4.10827, stops instead of continuing forever. Option D, 4.10810827…, repeats the wrong part. Exam cue: identify exactly which digits the bar covers before writing the expansion.

In lowest form, a decimal terminates only when \(q=2^m5^n\). An even denominator alone is not enough, since \(6=2\cdot3\) also contains 3. Exam tip: reduce the fraction to lowest form first.

( \frac{11}{17}=0.647058\ldots ), which is less than (0.6471). View all numbers in decimal form for comparison.

Direct answer: option B, 4. Compare the decimal numbers from left to right. In 0.83a2 and 0.8352, the digits 8 in the tenths place and 3 in the hundredths place are equal. The next digit is the thousandths digit: it is a in the first number and 5 in the second. For 0.83a2<0.8352, we need a<5. Since a is a digit, its possible values are 0,1,2,3,4; the greatest is 4. Option A, 3, satisfies the inequality but is not the greatest possible digit. Option B, 4, is less than 5 and is the greatest valid value. Option C, 5, would make both numbers 0.8352, giving equality, not a strict less-than sign. Option D, 6, would make the first number larger than 0.8352. Compare decimal places in order and stop at the first unequal digit.

The number of zeros between (1)'s keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(1024) is a power of (2), so the decimal terminates and ( \frac{77}{1024}=0.0751953125 ). Convert a power-of-(2) denominator into a power of (10).

(0.00046875=\frac{46875}{100000000}=\frac{3}{6400}). Always write the final answer in simplest form.

To classify the decimal expansion of 47/72, first check whether the fraction is already in lowest form. The numerator 47 is prime and does not divide 72, so no common factor can be cancelled. Now factor the denominator: 72 = 2^3 × 3^2.

A fraction in lowest form has a terminating decimal only if its denominator has no prime factors other than 2 and 5. Although 72 contains a factor 2, it also contains the factor 3, so the required condition fails. The decimal therefore continues indefinitely but repeats a pattern, making it non-terminating recurring. Thus option B is correct. It is not non-recurring, because every rational number has a terminating or recurring decimal expansion.

The first (4) is in tenths and the last (4) is in millionths. Zero-value places need not be written separately.

Multiplying both numbers by (10000) gives (7040\div16=440). Removing decimals makes division easier.

Writing equal decimal places gives (0.281250000+0.003906250+0.000078125=0.285234375). Align decimal points while adding.

In 13.24999\ldots, infinitely many 9s occur after 13.24. An infinite tail of 9s increases the preceding place value by 1; for example, 0.00999\ldots = 0.01. Hence, 13.24999\ldots = 13.25. Option 13.249 is only a truncated value and is not equal to the given decimal. Exam tip: Replace an infinite string of 9s by adding 1 to the digit immediately before it.

( \frac{31}{64}=0.484375 ) and ( \frac{17}{35}=0.485714\ldots ), so (0.4846) lies between them. Convert the boundaries into decimals.

The zero gaps after successive 1s are 1, 2, 3, ..., so no fixed digit block repeats. Hence it is non-terminating and non-recurring, therefore irrational. Exam tip: look for a repeating block, not merely repeated digits.

(x=3.90625\div100000=0.0000390625). On dividing by (100000), the decimal moves five places left.

In 32.008009, the digits after the decimal point occupy the tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths places, respectively. The digit 9 is in the sixth place, the millionths place. Therefore, its place value is \(9 \times \frac{1}{1000000}=\frac{9}{1000000}\). Option C, \(\frac{9}{100000}\), represents the fifth-place value, not the sixth-place value. Exam tip: Count decimal places from left to right, beginning with tenths.

Write all the numbers up to four decimal places for comparison: \(7.0070, 7.0707, 7.0077, 7.7000\). Comparing digits after the decimal point from left to right gives \(7.0070<7.0077<7.0707<7.7000\). Hence, option A is correct. In option D, \(7.0707\) and \(7.0077\) are placed in the wrong order because, at the tenths place, \(0<7\). Exam tip: Add trailing zeros when needed to make the decimal places equal before comparing decimals.

(16000\times625=10000000), so ( \frac{173}{16000}=0.0108125 ). Convert the denominator into a power of (10) to get the exact decimal.