Decimal representation

The governing concept is recurring decimal notation. In 1.232323..., the digits after the decimal point are 2, 3, 2, 3, 2, 3, and so on. The two-digit block 23 repeats continuously, so the recurring part is 23 and option B is correct. The initial digit 1 is the whole-number part, not a repeating decimal block. Although the sequence also contains 32 if read from the second digit onward, 32 is not aligned with the repeated cycle beginning immediately after the decimal point; writing the decimal as 1.232323... clearly shows the period 23. The block 123 is not repeated at all. Identifying the smallest block that reproduces the decimal indefinitely is the standard way to determine the repetend.

It has no fixed repeating pattern and it does not terminate. Such a decimal is non-terminating non-recurring.

A terminating decimal has a finite number of digits after the decimal point. A rational number can have terminating or non-terminating recurring decimal expansion.

\(0.272727\ldots\) is recurring, and \(0.272727\ldots=\frac{27}{99}=\frac{3}{11}\), so it is rational. \(\sqrt{2}\) and \(\pi\) are irrational. Exam tip: every recurring decimal represents a rational number.

Dividing 2 by 3 gives 0.666…, where the digit 6 repeats indefinitely. Therefore, \(\frac{2}{3}=0.666\ldots\). Option B is incorrect because 0.6 is only a one-decimal-place approximation, not the exact value. Exam tip: Remember that \(\frac{1}{3}=0.333\ldots\); doubling it gives \(\frac{2}{3}=0.666\ldots\).

( \frac{5}{8}=\frac{625}{1000}=0.625 ). A denominator of (8) can often be converted to (1000).

(8) is the second digit after the decimal point, so it is in the hundredths place. Count digits after the decimal in order to identify place value.

The correct answer is 0.50 because adding a zero at the right end of a decimal does not change its value. Hence, 0.5 = 0.50. Option 0.05 is one-tenth of 0.5, so it is not equal to it. Exam tip: Zeros added or removed at the end of a decimal do not change its value.

To compare decimals, compare digits from left to right, giving equal place value to each number by adding zeros when necessary. Write the numbers as 0.7500, 0.7050, 0.5700, and 0.7501. The first three digits of 0.7500 and 0.7501 are equal, but at the fourth decimal place, 1 is greater than 0. Therefore 0.7501 is greater than 0.7500, so choice D is correct.

Notice that 0.75 and 0.7500 have exactly the same value; adding zeros at the end does not change a decimal. However, 0.7501 has an additional positive amount, one ten-thousandth, so it is slightly larger. The number 0.705 is smaller because its hundredths digit is 0, and 0.57 is smaller still because its tenths digit is 5 rather than 7.

(2.09) is smaller than (2.099) and the other options. In decimal comparison, missing places can be treated as zeros.

\(0.\overline{3}\) is non-terminating but recurring, and \(0.\overline{3}=\frac{1}{3}\); hence it is rational. \(\sqrt{2}\) and \(\pi\) are non-terminating, non-recurring irrational numbers. Exam tip: every recurring decimal is rational.

In (0.04), (4) is in the hundredths place, so it is ( \frac{4}{100} ). For two decimal places, use denominator (100).

In 2.35, 2 is the whole-number part and 35 represents thirty-five hundredths, so the mixed fraction is \(2+\frac{35}{100}=2\frac{35}{100}\). Option B has denominator 10 and gives 5.5, while option C reverses the whole-number and fractional parts. Exam tip: when there are two digits after the decimal point, write the decimal part over 100.

(7) is in the third place after the decimal, so it is in the thousandths place. Zeros are also important while counting places.

(25=5^2), so the denominator has only the factor (5). Such a fraction has a terminating decimal expansion.

(40=2^3\times5), so the decimal will terminate. If the denominator has only (2) and (5), the decimal terminates.

(6=2\times3) includes (3), so the decimal will not terminate. Since it is rational, it will be non-terminating recurring.

(7) is neither (2) nor (5), so the decimal will not terminate. Since it is rational, it will be non-terminating recurring.

Since 0.375 has three digits after the decimal point, it can be written as \(\frac{375}{1000}\). Dividing the numerator and denominator by 125 gives \(\frac{375}{1000}=\frac{3}{8}\), which is the simplest form. In option B, the decimal digits have been handled incorrectly, while options C and D are not equal to 0.375. Exam tip: use a denominator of 1 followed by as many zeros as there are decimal places, then reduce the fraction.

The governing concept is place value in decimal representation. A denominator of 100 means that the numerator is measured in hundredths, so two digits must be placed to the right of the decimal point. Therefore 11/100 = 0.11. Option B is correct. The value 1.1 is eleven tenths, or 110/100, so it is ten times too large. The value 0.011 represents eleven thousandths, or 11/1000, so it is ten times too small. The number 11.00 is simply 11, which is much larger than 0.11. Another useful check is multiplication: 0.11 × 100 = 11, confirming the fraction exactly. Zeros may be added at the left when needed, but changing the number of decimal places changes the denominator and therefore the value.