Decimal representation

Correct answer: A. Terminating decimal

Explanation: 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.

Correct answer: C. 0.6666\ldots

Explanation: 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.

Correct answer: C. Rational

Explanation: 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.

Correct answer: B. \(\frac{13}{40}\)

Explanation: 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.

Correct answer: B. \(\frac{1}{2}\)

Explanation: 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.

Correct answer: B. Non-terminating repeating

Explanation: 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.

Correct answer: A. (0.0688)

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

Correct answer: B. ( \frac{1}{320} )

Explanation: 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.

Correct answer: C. Non-terminating recurring

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

Correct answer: A. Terminating

Explanation: 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.

Correct answer: B. \(0.12\overline{03}\)

Explanation: 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.

Correct answer: A. (0.3181818\ldots)

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

Correct answer: A. (0.772727\ldots)

Explanation: 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.

Correct answer: B. (0.417)

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

Correct answer: C. (7)

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

Correct answer: C. Non-terminating non-recurring

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

Correct answer: A. (0.109375)

Explanation: (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).

Correct answer: C. Terminating because the simplest form is ( \frac{2}{5} )

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

Correct answer: B. (2+\frac{5}{10}+\frac{5}{10000})

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

Correct answer: B. (250)

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

Correct answer: A. 2.4665

Explanation: 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.

Correct answer: A. (0.044625)

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

Correct answer: C. (0.999\ldots=1)

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

Correct answer: C. (0.554)

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

Correct answer: B. 4

Explanation: 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.