Decimal representation

(2.155) is greater than (2.150) and less than (2.160). Thinking on a number line helps identify numbers between two decimals.

The governing idea is comparison of decimals using place value and equivalent notation. Express the boundaries as 0.400 and 0.500 so that all relevant places can be compared. The number 0.405 has the same whole-number and tenths digits as the lower boundary, but its thousandths digit makes it slightly larger than 0.400. It is still less than 0.500, so 0.400 < 0.405 < 0.500. Hence option B is correct. The value 0.39 is below 0.4, while 0.50 is exactly equal to the upper boundary and therefore is not strictly less than 0.5. The value 0.504 is greater than 0.5. Writing a trailing zero in 0.4 and 0.5 does not alter their values; it simply allows a fair digit-by-digit comparison.

In the decimal 0.272727…, the block 27 repeats continuously. Therefore, the bar is placed over the complete repeating block: \(0.\overline{27}\). Option \(0.2\overline{7}\) is incorrect because it represents 0.27777…. Exam tip: first identify the shortest block of digits that repeats.

In 5.1333..., the digit 1 occurs once after the decimal point, and then 3 repeats forever. Therefore, the smallest repeating part is \(3\). Although \(33\) may appear as a repeated group, the repeating block is taken in its shortest form, which is \(3\). Exam tip: identify the shortest digit or group of digits that repeats continuously in a recurring decimal.

In a decimal expansion, the recurring part is the block of digits that repeats indefinitely in the same order. It is important to separate any nonrepeating digits at the beginning from the repeating block. In 2.4585858…, the digits after the decimal point begin 4, 5, 8, 5, 8, 5, 8 and so on.

The digits 45 occur only once at the start of the decimal part. After them, the two-digit block 58 repeats continuously: 58, 58, 58, … . Therefore the recurring part is 58, which is choice B. The block 858 is not the basic repeating block, because it overlaps the repeated pattern and does not represent the shortest repeating unit. The initial 45 is nonrecurring.

A reduced fraction terminates only if its denominator has prime factors 2 and/or 5. Here \(30=2\times3\times5\), so \(\frac{7}{30}=0.2333\ldots\) is recurring. Exam tip: factorise the reduced denominator first.

(30=2\times3\times5) also has (3), so the decimal will not terminate. Since it is rational, it will be non-terminating recurring.

\(0.375=\frac{375}{1000}=\frac{3}{8}\), so it is a terminating rational decimal. \(\sqrt{2}\) and \(\pi\) are irrational. Exam tip: convert a terminating decimal into a fraction to verify its type.

\(\frac{7}{40}\) is correct because \(40=2^3\times5\). A fraction in lowest form has a terminating decimal only when its denominator has no prime factors other than 2 and 5. The denominators 30, 45, and 66 contain other primes. Exam tip: factorise the denominator first.

\(0.125=\frac{125}{1000}=\frac{1}{8}\), so it can be written as a ratio of integers. Its decimal expansion terminates, hence it is rational. A decimal point or being non-integral does not make a number irrational. Exam tip: terminating or recurring decimals are rational.

(12=2^2\times3) has (3), so the decimal will not terminate. Since it is rational, it will be recurring.

The governing concept is converting a terminating decimal to a fraction and reducing it to lowest terms. Because 0.625 has three digits after the decimal point, write it as 625/1000. The greatest common divisor of 625 and 1000 is 125. Dividing numerator and denominator by 125 gives 625 ÷ 125 = 5 and 1000 ÷ 125 = 8, so 0.625 = 5/8. Therefore option A is correct. Although 25/40 is equivalent to 5/8, it is not in simplified form, so option C does not satisfy the wording. The fraction 3/8 equals 0.375, not 0.625. The expression 625/100 equals 6.25 and also has not used the correct denominator for three decimal places. The final fraction 5/8 cannot be reduced further because 5 and 8 have no common factor greater than 1.

The decimal \(0.0125\) has four digits after the decimal point, so it can be written as \(\frac{125}{10000}\). Dividing the numerator and denominator by 125 gives \(\frac{125}{10000}=\frac{1}{80}\). Therefore, option B is correct. Exam tip: For a terminating decimal, use 1 followed by as many zeros as there are decimal places, then reduce the fraction.

(0.75\times100=75), so it becomes (75%). To convert a decimal into a percentage, multiply by (100).

The correct answer is 35 because \(3.5 \times 10 = 35\). When a number is multiplied by 10, its decimal point shifts one place to the right: 3.5 → 35. Option B, 3.50, is equal to 3.5, so it does not show the product. Exam tip: When multiplying by 10, 100, or 1000, shift the decimal point 1, 2, or 3 places to the right, respectively.

The correct answer is 4.27 because dividing by 10 shifts the decimal point one place to the left: \(42.7 \div 10 = 4.27\). The distractor 0.427 would result from dividing by 100, not by 10. Exam tip: When dividing by 10, 100, or 1000, move the decimal point 1, 2, or 3 places to the left, respectively.

We can write 0.4 as 0.40 because adding a zero at the end of a decimal does not change its value. Therefore, 0.36 + 0.40 = 0.76. Option 0.396 incorrectly places digits together instead of adding according to place value. Exam tip: Always align the decimal points before adding decimal numbers.

0.5 can be written as 0.50 because adding a zero to the right of a decimal does not change its value. Thus, 1.25 - 0.50 = 0.75. The option 0.80 may result from incorrect subtraction of decimal digits. Exam tip: Align the decimal points before subtracting decimal numbers.

(0.05=0.050), which is greater than (0.005). The position of zero is very important in decimal comparison.

The whole-number part of a decimal is the part before the decimal point. In 15.08, the decimal point separates 15 from 08. Therefore the whole-number part is 15, making choice A correct. The digits 0 and 8 are after the decimal point and describe the fractional part, which is 0.08 or eight hundredths.

Place value makes the distinction clear: 1 is in the tens place and 5 is in the ones place, so they form the integer 15. The zero in the tenths place and the 8 in the hundredths place together form the decimal portion. Writing 08 alone does not identify the whole-number part, and writing 1508 would remove the decimal point and change the value. Thus 15 is the required answer.