Operations on Numbers Questions and Answers

Q: If \(1^2 + 2^2 + 3^2 + ... + 10^2 = 385\), then the value of \(2^2 + 4^2 + 6^2 + ... + 20^2\) is:

Let the required sum be S.\(S = 2^2 + 4^2 + 6^2 + ... + 20^2\).\(S = (2 \times 1)^2 + (2 \times 2)^2 + (2 \times 3)^2 + ... + (2 \times 10)^2\).\(S = 2^2(1^2) + 2^2(2^2) + 2^2(3^2) + ... + 2^2(10^2)\).\(S = 2^2 (1^2 + 2^2 + 3^2 + ... + 10^2)\).\(S = 4 \times (\text{given sum})\).\(S = 4 \times 385 = 1540\).

Q: What is the sum of all prime numbers between 60 and 80?

We need to identify the prime numbers between 60 and 80.Prime numbers are: 61, 67, 71, 73, 79.Sum = \(61 + 67 + 71 + 73 + 79\).Sum = \((60+1) + (60+7) + (70+1) + (70+3) + (70+9) = 120 + 210 + (1+7+1+3+9) = 330 + 21 = 351\).

Q: A number when divided by 296 gives a remainder of 75. When the same number is divided by 37, the remainder is:

Let the number be N. N = 296k + 75.We need to find the remainder when N is divided by 37.First, check if 296 is divisible by 37. \(296 = 37 \times 8\).Since the first divisor is a multiple of the second divisor, we can simply divide the first remainder by the second divisor.Remainder = \(75 \div 37\).\(75 = 2 \times 37 + 1\).The remainder is 1.

Q: A number whose fifth part increased by 4 is equal to its fourth part diminished by 10. The number is:

Let the number be x.The equation is \(\frac{x}{5} + 4 = \frac{x}{4} - 10\).\(4 + 10 = \frac{x}{4} - \frac{x}{5}\).\(14 = \frac{5x - 4x}{20}\).\(14 = \frac{x}{20}\).\(x = 14 \times 20 = 280\).

Q: What is the value of \(\sqrt[3]{4 \frac{12}{125}}\)?

First, convert the mixed fraction to an improper fraction.\(4 \frac{12}{125} = \frac{4 \times 125 + 12}{125} = \frac{500+12}{125} = \frac{512}{125}\).Now find the cube root: \(\sqrt[3]{\frac{512}{125}} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}}\).We know \(8^3 = 512\) and \(5^3 = 125\).So the value is \(\frac{8}{5} = 1.6\).

Q: If the product of three consecutive integers is 120, then the sum of the integers is:

Let the consecutive integers be \(n-1, n, n+1\).We need to find three consecutive numbers whose product is 120. We can estimate the cube root of 120. \(5^3=125\), so the numbers should be around 5.Let's try 4, 5, and 6.\(4 \times 5 \times 6 = 20 \times 6 = 120\). This is correct.The integers are 4, 5, and 6.Their sum is \(4 + 5 + 6 = 15\).

Q: What is the next number in the sequence: 1, 4, 9, 16, 25, ...?

The terms are the perfect squares.\[ 1^{2},\ 2^{2},\ 3^{2},\ 4^{2},\ 5^{2} \]\[ \text{next} = 6^{2} = 36 \]The 49 on offer is \( 7^{2} \), one term too far.

Q: What is the equivalent of 0.75 as a fraction?

\(0.75\) can be written as \(\frac{75}{100}\).To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 25.\(75 \div 25 = 3\).\(100 \div 25 = 4\).So, the fraction is \(\frac{3}{4}\).

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91.

If \(1^2 + 2^2 + 3^2 + ... + 10^2 = 385\), then the value of \(2^2 + 4^2 + 6^2 + ... + 20^2\) is:

A.770
B.1155
C.1540
D.3850

Answer: C

Let the required sum be S.

\(S = 2^2 + 4^2 + 6^2 + ... + 20^2\).

\(S = (2 \times 1)^2 + (2 \times 2)^2 + (2 \times 3)^2 + ... + (2 \times 10)^2\).

\(S = 2^2(1^2) + 2^2(2^2) + 2^2(3^2) + ... + 2^2(10^2)\).

\(S = 2^2 (1^2 + 2^2 + 3^2 + ... + 10^2)\).

\(S = 4 \times (\text{given sum})\).

\(S = 4 \times 385 = 1540\).

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92.

What is the sum of all prime numbers between 60 and 80?

A.351
B.350
C.349
D.345

Answer: A

We need to identify the prime numbers between 60 and 80.

Prime numbers are: 61, 67, 71, 73, 79.

Sum = \(61 + 67 + 71 + 73 + 79\).

Sum = \((60+1) + (60+7) + (70+1) + (70+3) + (70+9) = 120 + 210 + (1+7+1+3+9) = 330 + 21 = 351\).

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93.

A number when divided by 296 gives a remainder of 75. When the same number is divided by 37, the remainder is:

A.1
B.2
C.8
D.11

Answer: A

Let the number be N. N = 296k + 75.

We need to find the remainder when N is divided by 37.

First, check if 296 is divisible by 37. \(296 = 37 \times 8\).

Since the first divisor is a multiple of the second divisor, we can simply divide the first remainder by the second divisor.

Remainder = \(75 \div 37\).

\(75 = 2 \times 37 + 1\).

The remainder is 1.

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94.

A number whose fifth part increased by 4 is equal to its fourth part diminished by 10. The number is:

A.240
B.260
C.280
D.300

Answer: C

Let the number be x.

The equation is \(\frac{x}{5} + 4 = \frac{x}{4} - 10\).

\(4 + 10 = \frac{x}{4} - \frac{x}{5}\).

\(14 = \frac{5x - 4x}{20}\).

\(14 = \frac{x}{20}\).

\(x = 14 \times 20 = 280\).

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95.

What is the value of \(\sqrt[3]{4 \frac{12}{125}}\)?

A.1.2
B.1.4
C.1.6
D.1.8

Answer: C

First, convert the mixed fraction to an improper fraction.

\(4 \frac{12}{125} = \frac{4 \times 125 + 12}{125} = \frac{500+12}{125} = \frac{512}{125}\).

Now find the cube root: \(\sqrt[3]{\frac{512}{125}} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}}\).

We know \(8^3 = 512\) and \(5^3 = 125\).

So the value is \(\frac{8}{5} = 1.6\).

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96.

Find the value of \(1/(2\times3) + 1/(3\times4) + 1/(4\times5) + ... + 1/(9\times10)\).

A.1/10
B.2/5
C.4/5
D.3/10

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 \(\frac{1}{2} - \frac{1}{10}\).

\(\frac{5-1}{10} = \frac{4}{10} = \frac{2}{5}\).

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97.

What will be the remainder when \(17^{200}\) is divided by 18?

A.1
B.2
C.16
D.17

Answer: A

We can use the remainder theorem.

\(17\) can be written as \((18 - 1)\).

So, we need to find the remainder of \((18-1)^{200} \div 18\).

Using the binomial expansion, every term except the last one will have 18 as a factor. The last term is \((-1)^{200}\).

\((-1)^{200} = 1\).

So the remainder is 1.

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98.

If the product of three consecutive integers is 120, then the sum of the integers is:

A.9
B.12
C.15
D.18

Answer: C

Let the consecutive integers be \(n-1, n, n+1\).

We need to find three consecutive numbers whose product is 120. We can estimate the cube root of 120. \(5^3=125\), so the numbers should be around 5.

Let's try 4, 5, and 6.

\(4 \times 5 \times 6 = 20 \times 6 = 120\). This is correct.

The integers are 4, 5, and 6.

Their sum is \(4 + 5 + 6 = 15\).

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99.

What is the next number in the sequence: 1, 4, 9, 16, 25, ...?

A.30
B.36
C.49
D.42

Answer: B

The terms are the perfect squares.

\[ 1^{2},\ 2^{2},\ 3^{2},\ 4^{2},\ 5^{2} \]

\[ \text{next} = 6^{2} = 36 \]

The 49 on offer is \( 7^{2} \), one term too far.

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100.

What is the equivalent of 0.75 as a fraction?

A.1/2
B.3/4
C.4/3
D.5/4

Answer: B

\(0.75\) can be written as \(\frac{75}{100}\).

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 25.

\(75 \div 25 = 3\).

\(100 \div 25 = 4\).

So, the fraction is \(\frac{3}{4}\).

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