Arithmetic Operations Questions and Answers

Answer: A

Convert the recurring decimals to fractions.

\[ 0.\overline{5} = \frac{5}{9}, \qquad 0.\overline{4} = \frac{4}{9} \]

\[ \frac{5}{9} + \frac{4}{9} = \frac{9}{9} = 1 \]

Adding the decimals directly gives \( 0.999\ldots \), which is exactly 1.

Answer: A

\[ S_{n} = \frac{n(n+1)}{2} \]

\[ S_{45} = \frac{45 \times 46}{2} = 45 \times 23 = 1035 \]

Answer: B

Let the two-digit number be 10x + y, where x is the ten's digit and y is the unit's digit.

The reversed number is 10y + x.

Their sum is \((10x + y) + (10y + x) = 11x + 11y\).

Sum = 11(x + y).

Since the sum is a multiple of 11, it will always be divisible by 11.

Answer: B

This is a telescoping series. Each term can be split using partial fractions: \(1/(n(n+1)) = 1/n - 1/(n+1)\).

The series becomes: \((\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) + ... + (\frac{1}{9} - \frac{1}{10})\).

All the intermediate terms cancel out.

We are left with \(\frac{1}{2} - \frac{1}{10} = \frac{5-1}{10} = \frac{4}{10} = \frac{2}{5}\).

Answer: C

Correct answer: \( \frac{4}{5} \)

Each denominator is a product of consecutive integers: \( 2 = 1 \times 2 \), \( 6 = 2 \times 3 \), \( 12 = 3 \times 4 \), \( 20 = 4 \times 5 \). Every term can therefore be split as \[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}, \] giving \[ \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \left(\frac{1}{4} - \frac{1}{5}\right). \] Everything between the ends cancels - a telescoping series - leaving \[ 1 - \frac{1}{5} = \frac{4}{5}. \] The general result is that the sum of the first \( n \) such terms is \( \dfrac{n}{n+1} \), and with \( n = 4 \) that is \( 4/5 \) directly. Adding the fractions by finding a common denominator of 60 gives \( 30 + 10 + 5 + 3 = 48 \), so \( 48/60 = 4/5 \) - correct, but slower.

Answer: A

For a number to be divisible by 8, the number formed by its last three digits must be divisible by 8.

The last three digits are 985.

We divide 985 by 8: \(985 \div 8 = 123\) with a remainder of 1.

To make it divisible by 8, we must subtract the remainder. So, we need to subtract 1.

Answer: B

The first multiple of 6 after 17 is 18 (6x3). The last multiple of 6 before 80 is 78 (6x13).

The numbers form an A.P.: 18, 24, ..., 78.

The number of terms can be found by the indices of the multiples: 3, 4, ..., 13.

Number of terms = (Last index - First index) + 1 = (13 - 3) + 1 = 11.

Answer: B

To multiply decimals, first multiply the numbers ignoring the decimal points: 3 × 124 = 372.

Now, count the total number of decimal places in the original numbers. 0.03 has 2 decimal places, and 0.0124 has 4 decimal places. The total is 2 + 4 = 6 decimal places.

Place the decimal point in the result so that it has 6 decimal places: 0.000372.

Answer: B

This question is indeterminate as there could be multiple combinations. Let x, y, z be the number of notes of 50, 20, 10. We have x+y+z=50 and 50x+20y+10z=1840. We have two equations and three variables. The question cannot be uniquely solved without more information. Let's assume there is a typo. If the question was: 'The number of Rs. 50 notes is 6 more than Rs. 20 notes'. Then x=y+6. z=50-x-y = 50-(y+6)-y=44-2y. 50(y+6)+20y+10(44-2y)=1840. 50y+300+20y+440-20y=1840. 50y+740=1840. 50y=1100. y=22. Then x=28, z=0. Total notes = 50. Let's check the value: 28*50 + 22*20 = 1400+440=1840. This works. So the number of Rs. 50 notes is 28. Let me adjust the question to include this constraint.

Answer: C

Let x, y, and z be the number of notes of Rs. 50, Rs. 20, and Rs. 10 respectively.

Equation 1 (Total notes): x + y + z = 50

Equation 2 (Total value): 50x + 20y + 10z = 1840

Equation 3 (Constraint): x = y + 6

Substitute (3) into (1): (y+6) + y + z = 50 => 2y + z = 44 => z = 44 - 2y.

Now substitute x and z into (2):

50(y+6) + 20y + 10(44-2y) = 1840

50y + 300 + 20y + 440 - 20y = 1840

50y + 740 = 1840

50y = 1100 => y = 22.

Now find x: x = y + 6 = 22 + 6 = 28.

The number of Rs. 50 notes is 28.