Decimal representation

The governing concept is recurring-decimal notation. A bar placed over a group of digits means that the entire group, not merely its last digit, repeats endlessly immediately after the decimal point. Therefore 0.̅12 means 0.12121212…, so option A is correct. Option B represents the terminating decimal 0.12, followed by zeros, and contains no repeated block 12. Option C begins with 11, so its decimal pattern is different. Option D begins with 012 and likewise does not match the indicated notation. Reading the complete block under the bar is essential. The same value can also be written as 12/99, which simplifies to 4/33, confirming that it is a recurring rational decimal.

In \(0.\overline{12}=0.121212\ldots\), the block 12 repeats endlessly, so it is non-terminating recurring. 0.625 and 2.75 terminate, while option C has no fixed repeating block. Exam tip: identify the shortest repeating block.

In (\frac{2}{11}), the denominator has (11), so the decimal will not terminate. Exam tip: check for denominator factors other than (2) and (5).

(4) is at the third place after the decimal point, so it is in the thousandths place. Exam tip: include zeros while counting places.

The direct answer is option D: \(0.75\). A fraction can be converted to a decimal by making the denominator 10, 100, or 1000, or by dividing the numerator by the denominator. Start with \(\frac34\). Multiply numerator and denominator by 25: \(\frac34=\frac{3\times25}{4\times25}=\frac{75}{100}=0.75\). Thus option D is correct. Another quick method is to remember \(\frac14=0.25\); three quarters are \(3\times0.25=0.75\). Option A, 0.34, is not \(\frac34\); multiplying by 100 gives 34%, not 75%. Option B, 0.43, similarly represents 43/100, not 75/100. Option C, 0.70, represents 70/100, which is less than three quarters. Option D, 0.75, represents 75/100 and reduces to \(3/4\) after dividing numerator and denominator by 25. Be careful not to read the digits 3 and 4 as the decimal 0.34. Memory cue: one quarter is 0.25, so three quarters are 0.75.

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.

The direct answer is option A, 0.34. A fraction means numerator divided by denominator, so \(17/50\) can be changed into an equivalent fraction with denominator 100. Multiply both numerator and denominator by 2: \(17/50=34/100\). One hundredths are written as two digits after the decimal point, so \(34/100=0.34\). Option A, 0.34, is correct. Option B, 0.17, would represent 17/100, not 17/50. Option C, 0.50, represents 1/2 and has no connection with the given numerator. Option D, 0.85, is the decimal value of 17/20, not 17/50. The safe exam method is to make the denominator 10, 100, or 1000 whenever possible.

(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.

The direct answer is B, 0.24. A fraction means division, so we calculate 6 divided by 25. To make the denominator 100, multiply both numerator and denominator by 4: \(\frac{6}{25}=\frac{6\times4}{25\times4}=\frac{24}{100}=0.24\). Therefore the decimal form is 0.24. Option A, 0.06, is incorrect because 6 hundredths is \(\frac{6}{100}\), not \(\frac{6}{25}\). Option B, 0.24, is correct because \(0.24=\frac{24}{100}=\frac{6}{25}\) after cancellation. Option C, 0.25, equals \(\frac{1}{4}\), so it is not the given fraction. Option D, 0.60, equals \(\frac{3}{5}\), which is much larger. A useful exam cue is to convert a denominator such as 25 into 100 whenever possible; multiplying by 4 makes the decimal immediate.

(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.

The direct answer is option A,
\(\frac{5}{8}\). The decimal has three digits after the decimal point, so first write
\(0.625=625/1000\). Now simplify by dividing numerator and denominator by 125:
\(625\div125=5\) and
\(1000\div125=8\). Hence
\(0.625=5/8\), and 5 and 8 have no common factor greater than 1, so the fraction is in simplest form. Option A is correct. Option B,
\(6/25\), equals 0.24, not 0.625. Option C,
\(25/6\), is greater than 1, while 0.625 is less than 1. Option D,
\(625/100\), equals 6.25 and is not simplified; the denominator should initially be 1000 for three decimal places. Memory cue: three decimal digits mean denominator 1000, then cancel.

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).

The direct answer is A: 0.\overline{4}. The decimal has 4 repeating forever: 0.444444... The bar is placed exactly above the digit or group that repeats. Here the repeating group is only 4, so the notation is 0.\overline{4}. Option A is correct because it shows one bar over 4. Option B, 0.4, means a terminating decimal and does not show that 4 continues forever. Option C places the bar over 44 but leaves another 4 outside it, which does not represent the given repeating pattern correctly. Option D, 4.\overline{0}, is a number greater than 4 and is not the same value. Remember: put the bar over the smallest complete repeating block.

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.