Decimal Fractions Questions and Answers

Answer: B

The value is 6004.

First divide: \(60.04 \div 0.02 = 6004 \div 2 = 3002\) (multiplying both numbers by 100 removes the decimals). Then:

\[ \begin{aligned} \ast \times 0.5 &= 3002 \\ \ast &= \frac{3002}{0.5} = 6004 \end{aligned} \]

Check: \(6004 \times 0.5 = 3002\). The options 60.04 and 600.4 misplace the decimal point, and 6004.4 adds a stray digit.

Answer: A

The descending order is 10.5, 1.50, 1.055, 1.05, 1.005, 0.155.

Compare the numbers place by place. 10.5 is the only number with a tens digit, so it is largest, and 0.155 the only one below 1, so it is smallest. Among the four numbers between 1 and 2, compare the tenths digit first: 1.50 has 5 tenths, and the others 0 tenths. Then compare hundredths: 1.055 and 1.05 both have 5 hundredths, and 1.055 is larger because of its extra thousandths; 1.005 has 0 hundredths.

\[ 1.50 \gt 1.055 \gt 1.050 \gt 1.005 \]

Option D puts 1.05 before 1.055, the error of treating the longer decimal as smaller; options B and C are not in descending order at all.

Answer: D

The value is 153000.

Remove the decimals. The numerator has 3 + 1 + 3 = 7 decimal places and the denominator 3 + 4 + 2 = 9, so the fraction equals the whole-number fraction times 10²:

\[ \begin{aligned} &\frac{125 \times 578 \times 729}{17 \times 81 \times 25} \times 100 \\ &= 5 \times 34 \times 9 \times 100 \\ &= 153000 \end{aligned} \]

using 125/25 = 5, 578/17 = 34 and 729/81 = 9. The other options are the same digits with the decimal point misplaced, which is what happens if the decimal places are not counted correctly.

Answer: C

The number in third place is 7.07.

Compare the numbers by writing them with the same number of decimal places: 7.070, 7.700, 7.707, 7.007 and 0.770. In ascending order:

0.770, 7.007, 7.070, 7.700, 7.707

The third is 7.07. The trap is reading 7.007 as larger than 7.07 because it has more digits; the tenths digit (0 against 0) and then the hundredths digit (0 against 7) decide the order.

Answer: B

The value is \(1\frac{1}{9}\).

A single repeating digit converts to that digit over 9: \(0.\overline{4} = \frac{4}{9}\) and \(0.\overline{6} = \frac{6}{9}\).

\[ \begin{aligned} 0.\overline{4} + 0.\overline{6} &= \frac{4}{9} + \frac{6}{9} \\ &= \frac{10}{9} = 1\frac{1}{9} \end{aligned} \]

Answering 1 treats the decimals as 0.4 and 0.6 and ignores the bars. \(1\frac{1}{10}\) comes from writing the recurring parts over 10 instead of 9, and \(1\frac{2}{9}\) is an addition slip.

Answer: A

The number to be added is 1.9789.

First find x, lining up the decimal places:

\[ \begin{aligned} x &= 1.011 + 10.11 - 12.101 + 0.1011 \\ &= 11.2221 - 12.101 = -0.8789 \\ 1.1 - (-0.8789) &= 1.9789 \end{aligned} \]

Because x is negative, more than 1.1 must be added. 0.2211 is what you get by treating x as +0.8789, and 0.3111 and 1.1311 come from misaligning the decimal places.

Answer: B

The value is \(6\frac{529}{990}\). The number 6.5\(\overline{34}\) has one non-repeating digit (5) and two repeating digits (34). For such a decimal the fraction is the number formed by all the digits after the point, minus the non-repeating part, over 990. \[0.5\overline{34} = \frac{534 - 5}{990} = \frac{529}{990}\] So the number is \(6\frac{529}{990}\). The denominator 900 would suit a number with two non-repeating digits and one repeating digit, and 999 a purely recurring three-digit block, so those options use the wrong denominators. The option with 531 comes from subtracting wrongly.

Answer: B

The value is 40000. Since \(\frac{1}{2.5} = 0.4\), write \(0.000025 = 2.5 \times 10^{-5}\). Then

4000 and 400 come from using the wrong power of ten, and 400000 comes from one power too many.

Another way: 0.000025 is 25 millionths, and \(\frac{1}{25 \times 10^{-6}} = \frac{10^{6}}{25} = 40000\). This agrees with the first method, which makes the answer 40000 and rules out 4000 and 400 (too small by factors of ten and a hundred) and 400000 (ten times too large).

Answer: A

The value is 611/4950. Let x = 0.12343434... Then 10000x = 1234.3434... and 100x = 12.3434..., and subtracting gives 9900x = 1222, so x = 1222/9900 = 611/4950. The fraction 611/9900 is half of that value, 1222/9999 would repeat the block 1222, and the bare 9900 printed in the third option is not a fraction of that value. Another way to see it: subtract the non-repeating part. The number is 0.12 followed by the repeating block 34. With x = 0.12343434..., 100x = 12.343434... and 10000x = 1234.343434..., so the repeating tail cancels when the first is subtracted from the second, leaving 9900x = 1222.

Answer: B

The value of x lies between 11 and 12.

Add the known numbers: 12345.6 + 123.456 + 1.23456 + 1234.56 = 13704.85056. Subtract this from 13716.08506.

\[ x=13716.08506-13704.85056=11.2345 \]

11.2345 lies between 11 and 12. The other options are 12 and 13 or negative ranges, which do not contain it.

A rough check helps: 12345.6 + 123.456 is about 12469, adding 1.23 and 1234.56 gives about 13704.8. The target is 13716.08, so x is about 11.2. This fits 'between 11 and 12' and not any of the other ranges, two of which are negative and cannot be right.