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Simplification Questions and Answers

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51.

To fill a tank, 25 buckets of water is required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-fifth of its present ?

Answer: C

Let the capacity of 1 bucket \(= x\)
Then, the capacity of tank\( = 25x\)
New capacity of bucket \(=\dfrac {2}{5}x\)
\(\therefore\)Required number of buckets
\(=\dfrac {25x}{\left( \dfrac {2x}{5} \right)}\\ =\left( 25x \times \dfrac {5}{2x} \right)\\ =\dfrac {125}{2}\\ =62.5\)

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52.

\(\dfrac{(723 + 1992)^2-(723-1992)^2}{723×1992}=\text{?}\)

Answer: A

\(\dfrac{(723+1992)^2-(723-1992)^2}{723×1992}\\=\dfrac{(a+b)^2-(a-b)^2}{ab}\quad \text{(where } a=723 \text{ and }b=1992)\\=\dfrac{a^2+2ab+b^2-\left(a^2-2ab+b^2\right)}{ab}\\=\dfrac{4ab}{4}=4\)

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53.

The price of 80 apples is equal to that of 120 oranges. The price of 60 apples and 75 oranges together is Rs. 1320. The total price of 25 apples and 40 oranges is:

Answer: B

Let price of one apple \(=a\)
price of one orange \(=b\)
Price of 80 apples = Price of 120 oranges
\(\Rightarrow 80a=120b\\ \Rightarrow 2a=3b\\ \Rightarrow b = \dfrac{2a}{3}~\cdots(1)\)
Price of 60 apples + Price of 75 oranges =Rs. 1320
\(\Rightarrow 60a+75b=1320\\ \Rightarrow 4a+5b=88\\ \Rightarrow 4a+\dfrac{5(2a)}{3}=88 \quad \text{[ Because substituted value of } b \text{ from }(1)]\\ \Rightarrow 12a+10a=88 \times 3\\ \Rightarrow 6a+5a=44 \times 3\\ \Rightarrow 11a=44 \times 3\\ \Rightarrow a=4 \times 3=12\)

\(b=\dfrac{2a}{3}=\dfrac{2 \times 12}{3}=8\)

Total price of 25 apples and 40 oranges
\(=25a+40b\\=25\times12+40\times8\\=300+320\\=620\)

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54.

There are 6 working days in a regular week and for each day, the working hours are 10. A man earns Rs. 2.10 per hour for regular work and Rs. 4.20 per hour for overtime. If he earns Rs. 525 in 4 weeks, how many hours did he work?

Answer: A

Regular working hours in 4 weeks
\(=4 \times 6 \times 10=240\) hours
Amount earned by working in these regular working hours
\(=240 \times 2.10= \text {Rs. } 504\)
Additional amount earned
\(=525-504=\text{Rs. }21\)
Hours he worked overtime
\(=\dfrac{21}{4.2}=\dfrac{210}{42}=5\) hours
Total hours he worked
\(=240+5=245\)

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55.

How many pieces of 0.85 meters can be cut from a rod 42.5 meters long?

Answer: C

\[ \frac{42.5}{0.85} = \frac{4250}{85} = 50 \]

Scaling both numbers by 100 clears the decimals before dividing.

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