Decimal representation

(\frac{3}{4}=\frac{75}{100}=0.75). Exam tip: you can also use (\frac{1}{4}=0.25) to solve quickly.

The governing concept is the separation of a decimal into its whole-number part and fractional decimal part. The decimal point divides these parts: digits on the left form the whole-number part, and digits on the right form the decimal part. In 12.03, the digits to the left are 12, so the whole-number part is 12 and option B is correct. The digits 03 on the right describe three hundredths, or 0.03, not the whole-number part. Option A selects only the final digit, while option D gives the decimal fraction. The leading zero in 03 has place-value significance for the hundredths position but does not change the fact that it lies after the decimal point.

(0.009=\frac{9}{1000}) because it has three decimal places. Exam tip: leading zeros after the decimal point also increase place count.

(3.4=3.40) because the final zero does not change the value. Exam tip: remove unnecessary zeros to identify equal decimals.

A decimal can be changed into a fraction by using a denominator based on the number of digits after the decimal point. In 8.75 there are two digits after the decimal, so it becomes \(\frac{875}{100}\). This fraction is then reduced by dividing numerator and denominator by their common factor 25, giving \(\frac{35}{4}\). Hence choice A is correct.

The value can also be checked by separating the whole and fractional parts: 8.75 is 8 plus 75 hundredths. Since \(8=\frac{32}{4}\) and \(0.75=\frac{3}{4}\), their sum is \(\frac{35}{4}\). Option B is equivalent in value but is not in simplest form. The other options do not equal 8.75, so they cannot be selected.

(0.124) is less than (0.125) because (4<5) at the thousandths place. Exam tip: compare digits place by place.

In (1.272727\ldots), (27) repeats, so it is not terminating. Exam tip: dots with repetition indicate a non-terminating recurring decimal.

(\frac{17}{50}=\frac{34}{100}=0.34). Exam tip: multiply denominator (50) by (2) to make (100).

(6) is in the thousandths place, so its place value is (\frac{6}{1000}). Exam tip: each zero shifts the place one step ahead.

A rational number has a terminating decimal expansion only when the denominator in lowest form has prime factors \(2\) and/or \(5\) only. Since \(20=2^2\times5\), \(\frac{7}{20}\) terminates. But \(30\) also has factor \(3\). Exam tip: reduce the fraction first.

In (0.1234567891011\ldots), no fixed repeating group is seen. Exam tip: such a decimal can be of irrational type.

The final zero in (0.20) does not change the value, so (0.2=0.20). Exam tip: remove trailing zeros before comparing.

(\frac{6}{25}=\frac{24}{100}=0.24). Exam tip: multiply denominator (25) by (4) to make (100).

(10.02) is greater than (10.01) because (2>1) at the hundredths place. Exam tip: when whole number parts are equal, compare decimal parts.

(0.625=\frac{625}{1000}=\frac{5}{8}). Exam tip: for three decimal places, first use denominator (1000).

For \(\frac{11}{40}\), the denominator is \(40=2^3\times5\). A rational number has a terminating decimal only when its reduced denominator has 2 and/or 5 as prime factors. The other denominators contain 3 or 7. Exam tip: reduce the fraction first.

It is irrational because its decimal expansion neither terminates nor repeats a fixed block of digits. A rational number has a terminating or recurring decimal expansion. Exam tip: check whether a fixed repeating block occurs; an ever-growing pattern is non-recurring.

A fraction with denominator 10 represents tenths. The numerator tells us how many tenths there are, so 7/10 means seven tenths. In decimal notation, tenths are written in the first place to the right of the decimal point. Therefore the fraction can be converted by dividing 7 by 10 or by shifting the decimal point in 7 one place to the left.

The calculation is \\(\frac{7}{10}=0.7\\). Writing 0.70 gives the same numerical value, but among the listed answers the direct decimal form 0.7 is choice D. The value 0.07 represents seven hundredths, or 7/100, not seven tenths. Similarly, 7.10 is greater than 7 and cannot represent this fraction. Thus choice D follows.

(0.125=\frac{125}{1000}=\frac{1}{8}). First write a decimal with denominator (10), (100), or (1000).

Since (4) repeats, a bar is placed over it. The bar is always written over the repeating part.