Operations on Numbers Questions and Answers

Q: Evaluate: \(8888 + 888 + 88 + 8\)

Line up the decimal columns and add.\[ 8888 + 888 + 88 + 8 = 9872 \]The units column gives \( 8+8+8+8 = 32 \), so the answer ends in 2 - enough on its own to pick it out of the options.

Q: Simplify: \(5005 - 5000 \div 10\)

Division comes before subtraction.\[ 5000 \div 10 = 500 \]\[ 5005 - 500 = 4505 \]Working left to right would give \( 5 \div 10 = 0.5 \) - the first option, and the reason the order of operations matters here.

Q: The sum of two numbers is 40 and their difference is 4. What is the ratio of the two numbers?

Let the two numbers be x and y.Given, x + y = 40 and x - y = 4.Adding the two equations: 2x = 44, so x = 22.Substituting x = 22 in the first equation: 22 + y = 40, so y = 18.The ratio of the numbers is x : y = 22 : 18. Simplifying this by dividing both by 2 gives 11 : 9.

Q: Find the value of \((51+52+53+...+100)\).

This is the sum of an arithmetic progression.Number of terms (n) = (100 - 51) + 1 = 50.Sum = \(\frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{50}{2} \times (51 + 100) = 25 \times 151 = 3775\).

Q: What is the smallest 5-digit number exactly divisible by 41?

The smallest 5-digit number is 10000.Divide 10000 by 41. \(10000 = 41 \times 243 + 37\).The remainder is 37.To find the number to be added to make it divisible by 41, we calculate (41 - 37) = 4.The required number is 10000 + 4 = 10004.

Q: If 1.5x = 0.04y, then what is the value of \(\frac{(y-x)}{(y+x)}\)?

Given 1.5x = 0.04y.This can be written as \(\frac{x}{y} = \frac{0.04}{1.5} = \frac{4}{150} = \frac{2}{75}\).Now we need to find \(\frac{(y-x)}{(y+x)}\). Divide the numerator and denominator by y: \(\frac{(1 - x/y)}{(1 + x/y)}\).Substitute the value of x/y: \(\frac{(1 - 2/75)}{(1 + 2/75)} = \frac{(73/75)}{(77/75)} = \frac{73}{77}\).

Q: The product of two numbers is 120 and the sum of their squares is 289. The sum of the two numbers is:

Let the numbers be x and y.Given, xy = 120 and \(x^2 + y^2 = 289\).We know that \((x+y)^2 = x^2 + y^2 + 2xy\).Substitute the given values: \((x+y)^2 = 289 + 2(120) = 289 + 240 = 529\).Therefore, x + y = \(\sqrt{529} = 23\).

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What is the value of the expression 101 + 102 + 103 + ... + 200?

A.15000
B.15050
C.15550
D.16050

Answer: B

The sum of an arithmetic progression is given by the formula \(S_n = \frac{n}{2}(a_1 + a_n)\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

Number of terms (n) = (Last Term - First Term) + 1 = (200 - 101) + 1 = 100.

First term (\(a_1\)) = 101.

Last term (\(a_n\)) = 200.

Sum = \(\frac{100}{2} \times (101 + 200) = 50 \times 301 = 15050\).

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Evaluate: \(8888 + 888 + 88 + 8\)

A.9872
B.9876
C.9862
D.9772

Answer: A

Line up the decimal columns and add.

\[ 8888 + 888 + 88 + 8 = 9872 \]

The units column gives \( 8+8+8+8 = 32 \), so the answer ends in 2 - enough on its own to pick it out of the options.

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What is the unit digit of \(7^{105}\)?

Answer: C

The cyclicity of the unit digit of powers of 7 is 4. The pattern is 7, 9, 3, 1.

Divide the power by 4 and find the remainder.

105 ÷ 4 gives a remainder of 1.

If the remainder is 1, the unit digit is the first in the cycle, which is 7.

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Simplify: \(5005 - 5000 \div 10\)

A.0.5
B.500
C.4505
D.4995

Answer: C

Division comes before subtraction.

\[ 5000 \div 10 = 500 \]

\[ 5005 - 500 = 4505 \]

Working left to right would give \( 5 \div 10 = 0.5 \) - the first option, and the reason the order of operations matters here.

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The sum of two numbers is 40 and their difference is 4. What is the ratio of the two numbers?

A.11:9
B.11:10
C.10:7
D.9:7

Answer: A

Let the two numbers be x and y.

Given, x + y = 40 and x - y = 4.

Adding the two equations: 2x = 44, so x = 22.

Substituting x = 22 in the first equation: 22 + y = 40, so y = 18.

The ratio of the numbers is x : y = 22 : 18. Simplifying this by dividing both by 2 gives 11 : 9.

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Find the value of \((51+52+53+...+100)\).

A.3775
B.3255
C.3575
D.3675

Answer: A

This is the sum of an arithmetic progression.

Number of terms (n) = (100 - 51) + 1 = 50.

Sum = \(\frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{50}{2} \times (51 + 100) = 25 \times 151 = 3775\).

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What is the smallest 5-digit number exactly divisible by 41?

A.10004
B.10000
C.10045
D.10025

Answer: A

The smallest 5-digit number is 10000.

Divide 10000 by 41.

\(10000 = 41 \times 243 + 37\).

The remainder is 37.

To find the number to be added to make it divisible by 41, we calculate (41 - 37) = 4.

The required number is 10000 + 4 = 10004.

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If 1.5x = 0.04y, then what is the value of \(\frac{(y-x)}{(y+x)}\)?

A.73/77
B.77/73
C.7.3/7.7
D.74/77

Answer: A

Given 1.5x = 0.04y.

This can be written as \(\frac{x}{y} = \frac{0.04}{1.5} = \frac{4}{150} = \frac{2}{75}\).

Now we need to find \(\frac{(y-x)}{(y+x)}\). Divide the numerator and denominator by y: \(\frac{(1 - x/y)}{(1 + x/y)}\).

Substitute the value of x/y: \(\frac{(1 - 2/75)}{(1 + 2/75)} = \frac{(73/75)}{(77/75)} = \frac{73}{77}\).

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The product of two numbers is 120 and the sum of their squares is 289. The sum of the two numbers is:

A.20
B.23
C.169
D.21

Answer: B

Let the numbers be x and y.

Given, xy = 120 and \(x^2 + y^2 = 289\).

We know that \((x+y)^2 = x^2 + y^2 + 2xy\).

Substitute the given values: \((x+y)^2 = 289 + 2(120) = 289 + 240 = 529\).

Therefore, x + y = \(\sqrt{529} = 23\).

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10.

What is the value of \((32 \times 12) + (17^2) - 95\)?

A.578
B.478
C.588
D.488

Answer: A

Follow the BODMAS rule. First, perform multiplication and find the square, then do addition and subtraction.

Step 1: \(32 \times 12 = 384\).

Step 2: \(17^2 = 289\).

Step 3: The expression becomes \(384 + 289 - 95\).

Step 4: \(384 + 289 = 673\).

Step 5: \(673 - 95 = 578\).

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Operations on Numbers Questions and Answers

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Operations on Numbers Questions and Answers

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