Operations on Numbers Questions and Answers

Answer: A

Let the number be x.

Two-fifth of the number is \(\frac{2}{5}x\).

The equation is \(x = \frac{2}{5}x + 75\).

\(x - \frac{2}{5}x = 75\).

\(\frac{3}{5}x = 75\).

\(x = 75 \times \frac{5}{3} = 25 \times 5 = 125\).

Answer: A

In the number 503535:

The first '3' is in the thousands place, so its place value is \(3 \times 1000 = 3000\).

The second '3' is in the tens place, so its place value is \(3 \times 10 = 30\).

The sum of the place values is \(3000 + 30 = 3030\).

Answer: A

Expand the square of the difference.

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

\[ 9 = 29 - 2ab \quad\Longrightarrow\quad 2ab = 20 \]

\[ ab = 10 \]

Answer: A

The required number is the HCF of the differences between the given numbers.

Differences are: \((91 - 43) = 48\), \((183 - 91) = 92\), and \((183 - 43) = 140\).

We need to find the HCF of 48, 92, and 140.

Prime factorization of 48 = \(2^4 \times 3\).

Prime factorization of 92 = \(2^2 \times 23\).

Prime factorization of 140 = \(2^2 \times 5 \times 7\).

The highest common factor is \(2^2 = 4\).

Answer: A

First, convert the mixed fractions to improper fractions.

\(2 \frac{1}{2} = \frac{5}{2}\)

\(3 \frac{1}{4} = \frac{13}{4}\)

\(1 \frac{1}{3} = \frac{4}{3}\)

The expression is \(\frac{5}{2} + \frac{13}{4} - \frac{4}{3}\).

The LCM of the denominators (2, 4, 3) is 12.

\(\frac{5 \times 6}{12} + \frac{13 \times 3}{12} - \frac{4 \times 4}{12} = \frac{30 + 39 - 16}{12} = \frac{53}{12}\).

Converting back to a mixed fraction: \(53 \div 12\) gives a quotient of 4 and a remainder of 5. So, \(4 \frac{5}{12}\).

Answer: B

Let the ten's digit be x and the unit's digit be y. Number = 10x + y.

Given: x + y = 9.

Given: (10x + y) + 27 = 10y + x (reversed number).

\(9x - 9y = -27\), which simplifies to \(x - y = -3\) or \(y - x = 3\).

We have two equations: x + y = 9 and y - x = 3.

Adding them: 2y = 12, so y = 6.

Substituting y=6 into x+y=9, we get x=3.

The number is 36.

Answer: C

We need to find the nearest perfect square greater than 2203.

We know \(40^2 = 1600\) and \(50^2 = 2500\). The number is between 40 and 50.

Let's try numbers ending in 3 or 7. \(47^2 = (50-3)^2 = 2500 - 300 + 9 = 2209\).

\(46^2 = (47-1)^2 = 2209 - 94 + 1 = 2116\).

The next perfect square after 2203 is 2209.

The number to be added is \(2209 - 2203 = 6\).

Answer: A

Align the decimal points and add.

\[ 0.600 + 0.660 + 0.066 + 6.606 = 7.932 \]

Padding each number to three decimal places first keeps the columns straight.

Answer: A

We are multiplying powers of 25 with different decimal places.

The expression can be written as \(25^1 \times (25 \times 10^{-1}) \times (25 \times 10^{-2}) \times (25 \times 10^{-3})\).

This is equal to \(25^4 \times 10^{-1-2-3} = 25^4 \times 10^{-6}\).

\(25^2 = 625\).

\(25^4 = (25^2)^2 = 625^2 = 390625\).

So, the result is \(390625 \times 10^{-6} = 0.390625\).

Answer: B

To solve this, express all terms with the same base, which is 2.

\(8 = 2^3\) and \(4 = 2^2\).

The equation becomes \((2^3)^x \times (2^2)^3 = 2^{12}\).

\(2^{3x} \times 2^{6} = 2^{12}\).

\(2^{3x+6} = 2^{12}\).

Equating the exponents: \(3x+6 = 12\).

\(3x = 6\).

\(x = 2\).