Decimal representation

rac{126}{224}=rac{9}{16} because both numerator and denominator are divisible by 14. In lowest form, the denominator is 16=2^4. A rational number rac{p}{q} has a terminating decimal expansion when, in lowest form, the prime factors of q are only 2 and/or 5. Hence, rac{9}{16}=0.5625 is terminating. A recurring decimal occurs when the denominator has a prime factor other than 2 or 5. Exam tip: reduce the fraction first, then factorise its denominator.

In option C, the digit 6 repeats indefinitely, so it is a recurring decimal. Every recurring decimal represents a rational number; for example, \(0.6666\ldots=\frac{2}{3}\). Options A and B have no fixed repeating block, and D is also not shown as a recurring decimal. Exam tip: Terminating and recurring decimals are always rational.

In the decimal 0.454545..., the block 45 repeats continuously, so it is a recurring decimal. Every recurring decimal is rational; here, 0.454545... = 45/99 = 5/11. It is neither an integer nor a natural number, while irrational numbers have non-terminating, non-recurring decimal expansions. Exam tip: A terminating or recurring decimal is always rational.

For \(\frac{13}{40}\), the denominator is \(40=2^3\times5\). A rational number in lowest terms has a terminating decimal expansion only when its denominator has no prime factors other than 2 and 5. \(\frac{7}{12}\) contains the factor 3 in its denominator, and \(\frac{11}{33}=\frac{1}{3}\), so they do not terminate. Exam tip: first reduce the fraction to lowest terms, then factorise the denominator.

In \(0.5000\ldots\), all digits after 5 are zeros, so its value remains \(0.5\). Now, \(0.5=\frac{5}{10}=\frac{1}{2}\); hence \(\frac{1}{2}\) is correct. The close distractor \(\frac{1}{5}\) equals \(0.2\), not \(0.5\). Exam tip: Zeros written at the end of a decimal do not change its value.

The decimal expansion of a rational number is either terminating or non-terminating repeating. Therefore, if its decimal expansion is infinite, a digit or a block of digits repeats, so option B is correct. A non-terminating non-repeating decimal represents an irrational number. Exam tip: For a rational number, an infinite decimal expansion must be repeating.

(625\times16=10000), so ( \frac{43}{625}=\frac{688}{10000}=0.0688 ). Making the denominator a power of (10) is the safest method.

The direct answer is B: \(1/320\). The number 0.003125 has six digits after the decimal point, so first write it as \(3125/1,000,000\). Now simplify. Dividing numerator and denominator by 3125 gives \(1/320\); equivalently, \(0.003125=125/40000=25/8000=5/1600=1/320\). Option A, \(1/32\), is ten times too large and would equal 0.03125. Option B is correct because \(1\div320=0.003125\). Option C, \(1/3200\), is ten times too small and equals 0.0003125. Option D, \(3125/100000\), is not the correct unsimplified fraction: six decimal places require denominator 1,000,000, not 100,000; it also equals 0.03125. A useful check is to count all decimal places carefully before simplifying.

The denominator has (7), a factor other than (2) and (5). So the rational number will have a non-terminating recurring decimal.

A rational number has a terminating decimal expansion when, after simplification, the denominator has no prime factors other than 2 and 5. First reduce the fraction by dividing 91 and 140 by their common factor 7: \\(\frac{91}{140}=\frac{13}{20}\\). The denominator 20 factors as \\(2^2\times5\\), so it contains only the allowed prime factors.

Indeed, \\(\frac{13}{20}=\frac{65}{100}=0.65\\), which ends after two decimal places. Therefore the expansion is terminating, and option A is correct. It is not non-terminating recurring because no factor such as 3, 7, or 11 remains in the simplified denominator. Simplifying before judging the decimal type is essential.

In the decimal expansion, after 12 the block 03 repeats as 03, 03, 03, … . Thus, 12 is the non-repeating part and the bar must be placed only over 03: \(0.12\overline{03}\). Option A incorrectly treats 1203 as the repeating block. Exam tip: Write the digits in order first, then identify the smallest block that repeats continuously.

The bar is only over (18), so after (3), (18) repeats. In bar notation, the barred part repeats continuously.

The direct answer is option A, 0.772727… . To convert 17/22 into a decimal, divide 17 by 22. Since 22 does not go into 17, write 0 and use 170; 22 goes 7 times, giving 154 and remainder 16. Bring down 0 to get 160; 22 goes 7 times, giving 154 and remainder 6. Bring down 0 to get 60; 22 goes 2 times, giving 44 and remainder 16. The remainder 16 has appeared before, so the digits 27 repeat: 17/22=0.772727… . Option A matches this. Option B, 0.727272…, is 8/11 and lacks the first 7. Option C, 0.818181…, equals 9/11, not 17/22. Option D does not correctly show the repeating pattern. Remember: when a remainder repeats in long division, the quotient digits from that point also repeat.

( \frac{5}{12}=0.41666\ldots ), which is less than (0.417). Think of all numbers in decimal form for comparison.

The tenths and hundredths digits are equal, so the thousandths digit must satisfy (a>6). Hence the smallest value is (7).

The number of zeros keeps changing, so there is no fixed recurring part. Such a decimal is non-terminating non-recurring.

(64\times15625=1000000), so ( \frac{7}{64}=0.109375 ). A denominator that is a power of (2) can be converted to a power of (10).

( \frac{18}{45}=\frac{2}{5} ), and (5) has only factor (5). It is necessary to simplify the fraction first.

The first (5) is in tenths and the last (5) is in ten-thousandths. Zero-value places need not be written separately.

Multiplying both by (10000) gives (1250\div5=250). Multiplying both parts of a division by the same number does not change the quotient.

Write 3.075 as 3.0750 so that both numbers have four decimal places. Then
\(3.0750-0.6085=2.4665\). Therefore, 2.4665 is correct. A value such as 2.4765 can result from misaligning place values. Exam tip: always align decimal points before subtracting decimals.

Writing equal decimal places gives (0.037500+0.006250+0.000875=0.044625). Align decimal points while adding.

(0.999\ldots=1) is a standard recurring decimal result. In exams, treat (0.\overline{9}) as equal to (1).

( \frac{11}{20}=0.55 ) and ( \frac{5}{9}=0.555\ldots ), so (0.554) lies between them. Convert the boundaries into decimals.

Both decimals are the same up to the hundredths place: 0.68. Therefore, compare the digits in the thousandths place. For 0.68b to be less than 0.684, b must be less than 4. Thus, b can be 0, 1, 2, or 3, giving 4 possible values. Option 3 would incorrectly exclude 0, which is also a digit. Exam tip: compare decimals from left to right until the first unequal digit appears.