Arithmetic Operations Questions and Answers

Answer: A

Correct answer: 94

In successive division each quotient becomes the next dividend, so work backwards from the last step.

Let the final quotient be \(c\). Then, reading the divisions in reverse:

\[b = 6c + 4, \qquad a = 5b + 3, \qquad N = 4a + 2\]

The least number comes from \(c = 0\), which gives \(b = 4\), then \(a = 5(4) + 3 = 23\), and

\[N = 4(23) + 2 = 94\]

Verifying the chain: \(94 \div 4\) gives quotient 23 remainder 2; \(23 \div 5\) gives quotient 4 remainder 3; \(4 \div 6\) gives quotient 0 remainder 4.

Answer: B

Use the identity for the difference of two squares of a sum and a difference.

\[ (a+b)^{2} - (a-b)^{2} = 4ab \]

\[ \frac{4ab}{ab} = 4 \]

The actual values 469 and 174 never have to be multiplied out - they cancel entirely.

Answer: C

A rational number is a number that can be expressed as a fraction p/q of two integers, where q is not zero. An irrational number cannot be expressed as such.

√2, √3, and √5 are irrational.

√4 = 2, which can be written as 2/1. Therefore, it is a rational number.

Answer: C

Clear the decimals by scaling both numbers by the same power of ten.

\[ \frac{0.000064}{0.016} = \frac{64}{16000} \]

\[ = 0.004 \]

The numerator has six decimal places and the denominator three, so the quotient has three.

Answer: A

Write each term as a power of ten less one.

\[ (100-1) + (1000-1) + (10{,}000-1) + (100{,}000-1) \]

\[ = 111{,}100 - 4 = 111{,}096 \]

Answer: C

A number is divisible by 25 if its last two digits form a number divisible by 25 (i.e., 00, 25, 50, or 75).

The last two digits of 1385 are 85.

The next multiple of 25 after 85 is 100. To get from 85 to 100, we need to add 15.

So, we must add 15 to the number (1385 + 15 = 1400).

Answer: A

Four consecutive odd numbers are symmetric about their average.

\[ \frac{112}{4} = 28 \]

The four numbers straddle 28 in steps of 2:

\[ 25,\ 27,\ 29,\ 31 \]

So the smallest is 25. Setting up \( x + (x+2) + (x+4) + (x+6) = 112 \) gives \( 4x + 12 = 112 \) and the same answer.

Answer: D

The numerator is in the form \(a^3+b^3\) and the denominator is in the form \(a^2-ab+b^2\). Let a=0.5 and b=0.3. Then \(a^3=0.125\) and \(b^3=0.027\). Denominator is \(a^2-ab+b^2 = 0.5^2 - (0.5)(0.3) + 0.3^2 = 0.25 - 0.15 + 0.09\). This matches.

We know that \(a^3+b^3 = (a+b)(a^2-ab+b^2)\).

So the expression simplifies to \(a+b\).

Value = 0.5 + 0.3 = 0.8.

Answer: A

First, express all bases as prime numbers.

\(4^{11} = (2^2)^{11} = 2^{22}\).

The expression is \(2^{22} \times 7^5 \times 11^2\).

The total number of prime factors is the sum of the exponents of the prime bases.

Total prime factors = 22 + 5 + 2 = 29.

Answer: C

Each packet weighs 300 g.

Convert everything to one unit and divide by the number of packets (three dozen = 36):

\[ \begin{aligned} \text{Total} &= 4.8 + 2.5 + 3.5 = 10.8\ \text{kg} \\ &= 10800\ \text{g} \\ \text{Each packet} &= \frac{10800}{36} = 300\ \text{g} \end{aligned} \]

The trap is the mixed units: the raisins are given in grams (2500 g = 2.5 kg). 250 g and 500 g do not come from any correct conversion, and 650 g would need a much larger total.