Decimal Fractions Questions and Answers

Answer: D

Work left to right, treating the subtraction as a negative term:

\[ 945.341 - 1042.792 = -97.451 \]

\[ -97.451 + 875.435 = 777.984 \]

\[ 777.984 + 31.025 = 809.009 \]

Grouping the positives first gives the same result: \( 945.341 + 875.435 + 31.025 = 1851.801 \) and \( 1851.801 - 1042.792 = 809.009 \).

Answer: C

Shifting the decimal point moves both numbers by a known power of ten, so the answer follows from the one given without redoing the division.

\[ \frac{29.94}{1.45} = \frac{2994 \div 100}{14.5 \div 10} = \frac{2994}{14.5} \times \frac{10}{100} = 172 \times \frac{1}{10} = 17.2 \]

The numerator shrank by a factor of 100 and the denominator by only 10, so the quotient shrinks by the ratio of those, a factor of 10.

Answer: D

Clear the decimal from the divisor by multiplying both numbers by the same power of ten, then divide.

(i) \[ \frac{35}{0.07} = \frac{3500}{7} = 500 \]

(ii) \[ \frac{2.5}{0.0005} = \frac{25000}{5} = 5000 \]

(iii) \[ \frac{136.09}{43.9} = \frac{1360.9}{439} \approx 3.1 \]

Dividing by a number smaller than 1 always makes the result larger, which is why the first two answers are so much bigger than the numbers they start from.

Answer: D

The three terms are the expansion of a squared difference, so no long multiplication is needed:

\[ a^{2} - 2ab + b^{2} = (a - b)^{2} \]

With \( a = 3.75 \) and \( b = 2.75 \),

\[ (3.75 - 2.75)^{2} = 1^{2} = 1 \]

Multiplying the three products out gives 14.0625, 20.625 and 7.5625, and \( 14.0625 - 20.625 + 7.5625 = 1 \) - the same answer for considerably more work.

Answer: D

Multiplying by a power of ten moves the decimal point right by as many places as the power has zeros.

(i) \[ 6.3204 \times 100 = 632.04 \]

(ii) \[ 0.069 \times 10000 = 690 \]

In the second the point moves four places but only three digits are available, so a zero is written in to fill the gap: 0.069 becomes 0690, that is 690.

Answer: A

Shifting the decimal point moves both numbers by a known power of ten, so the answer follows from the one given without redoing the division.

\[ \frac{29.94}{1.45} = \frac{2994 \div 100}{14.5 \div 10} = \frac{2994}{14.5} \times \frac{10}{100} = 172 \times \frac{1}{10} = 17.2 \]

The numerator shrank by a factor of 100 and the denominator by only 10, so the quotient shrinks by the ratio of those, a factor of 10.

Answer: B

A vulgar fraction is a fraction of two whole numbers. Write the decimal over the power of ten matching its number of decimal places, then cancel.

\[ 0.75 = \frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \]

Two decimal places give a denominator of \( 10^{2} = 100 \), and the highest common factor of 75 and 100 is 25.

The distractors are easy to rule out by dividing: \( \tfrac{3}{5} = 0.6 \), \( \tfrac{3}{2} = 1.5 \) and \( \tfrac{3}{7} \approx 0.4286 \).

Answer: B

Suppose commodity X will cost 40 paise more than Y after z years.
Then,
\((4.20 + 0.40z) - (6.30 + 0.15z) = 0.40\\ \Rightarrow 0.25z = 0.40 + 2.10\\ \Rightarrow z= \dfrac {2.50}{0.25} = \dfrac {250}{25} = 10\)

X will cost 40 paise more than Y 10 years after 2001 i.e., 2011

Answer: A

\((a + b)^2 = a^2 + 2ab + b^2\)
\(12.86 \times 12.86 + 12.86 \times p + 0.14 \times 0.14\)
\(= (12.86)^2 + 12.86 \times p + (0.14)^2\)
This can be written as,
\((12.86 + 0.14)^2 = 13^2\)
If, \(12.86 \times p = 2 \times 12.86 \times 0.14\)
i.e., if \(p = 2 \times 0.14 = 0.28\)

Answer: B

Again the numerator is a difference of two squares, so the whole expression collapses to the sum of the two numbers.

\[ \frac{a^{2}-b^{2}}{a-b} = \frac{(a+b)(a-b)}{a-b} = a+b \]

\[ \frac{(2.39)^{2}-(1.61)^{2}}{2.39-1.61} = 2.39 + 1.61 = 4 \]

The awkward decimals are deliberate: they punish anyone who squares first (\( 5.7121 - 2.5921 = 3.12 \), then \( 3.12 \div 0.78 = 4 \)) and reward spotting the identity.