Decimal representation

(2.125=\frac{2125}{1000}=\frac{17}{8}). A mixed decimal can also be converted into an improper fraction.

After the decimal point, the first place is tenths, the second is hundredths, and the third is thousandths. In 0.009, 9 is the third digit after the decimal point, so it is in the thousandths place. The hundredths digit in 0.009 is 0, not 9. Exam tip: Count decimal places in order: tenths, hundredths, then thousandths.

(4) is in the first place after the decimal point, so its place value is ( \frac{4}{10} ). The first decimal place is tenths.

In 8.032, the digit 3 is in the second place to the right of the decimal point. This is the hundredths place, so its place value is \(3 \times \frac{1}{100}=\frac{3}{100}\). Remember that 3 is the face value, whereas \(\frac{3}{100}\) is its place value.

After the decimal point, the places represent tenths, hundredths, thousandths and ten-thousandths. In 0.0008, 8 is in the fourth place after the decimal point, so its place value is \(\frac{8}{10000}\). Exam tip: Count decimal places from immediately after the decimal point.

In 4.73, 4 is in the ones place, 7 is in the tenths place, and 3 is in the hundredths place. Therefore, its expanded form is \(4+\frac{7}{10}+\frac{3}{100}\). In option B, the place values of 7 and 3 are interchanged. Exam tip: the first digit after the decimal has denominator 10, and the second has denominator 100.

In 0.508, 5 is in the tenths place, 0 is in the hundredths place, and 8 is in the thousandths place. Therefore, its expanded form is \(\frac{5}{10}+\frac{0}{100}+\frac{8}{1000}\). Option A omits the tenths term, while option C incorrectly places 8 in the hundredths place. Exam tip: the first, second, and third digits after the decimal point represent tenths, hundredths, and thousandths, respectively.

Adding a zero to the right end of 0.08 does not change its value, so 0.08 = 0.080. In 0.008, the digit 8 is in the third decimal place, so its value is different. Exam tip: Zeros added only at the end of a decimal do not change its value.

Zeros at the end of the decimal part do not change the value of a number. Thus, 6.500 = 6.5, so option B is correct. In 6.05 and 6.005, the zeros are in different places, so their values are not equal to 6.500. Exam tip: Only trailing zeros in a decimal can be added or removed without changing its value.

(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.

Direct answer: Option C, 75%. A percentage means “per hundred.” To convert a decimal into a percentage, multiply it by 100 and attach the percent sign. Therefore, \(0.75\times100=75\), so \(0.75=75\%\). Another way is to write 0.75 as \(75/100\), which directly means 75 percent. Option C is correct. Option A, 7.5%, would result from multiplying by 10, not by 100, so it is ten times too small. Option B, 0.75%, would mean \(0.75/100=0.0075\), not 0.75; it incorrectly treats the decimal as if it were already a percent. Option D, 750%, would result from multiplying by 1000 and is ten times too large. A useful reasonableness check is that 0.75 is less than 1, so its percentage must be less than 100%; 75% fits this, while 750% does not. Memory cue: decimal to percent means move the decimal point two places right and write %.

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.