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Decimal Fractions Questions and Answers

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31.

If 2994/14.5 = 172, then 29.94/1.45 = ?

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.

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32.

What will be the vulgar fraction of 0.75 ?

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 \( \tfrac{3}{7} \approx 0.4286 \).

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33.

The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2001, the price of commodity X was Rs. 4.20 and that of Y was Rs. 6.30, in which year commodity X will cost 40 paise more than the commodity Y ?

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

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34.

The expression \((12.86 \times 12.86 + 12.86 \times p + 0.14 \times 0.14)\) will be a perfect square for \(p\) equal to:

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 \(p = 2 \times 0.14 = 0.28\)

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35.

Evaluate : \(\dfrac{(2.39)^2 - (1.61)^2}{2.39 - 1.61}\)

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.

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36.

Arrange the fractions in ascending order.

Answer: D

\(\dfrac {5}{8} = .625\)
\(\dfrac {7}{12} = .5833\)
\(\dfrac {3}{4} = .75\)
\(\dfrac {13}{16} = .8125\)
So order will be
\(\dfrac {7}{12} < \dfrac {5}{8} < \dfrac {3}{4}< \dfrac {13}{16}\)

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