Simplify: \(1 \div [1 \div \{1 \div (1 \div 1)\}]\).
Answer: A
Work outward from the innermost bracket.
\[ 1 \div 1 = 1 \quad\Longrightarrow\quad 1 \div 1 = 1 \quad\Longrightarrow\quad 1 \div 1 = 1 \]
Every division returns 1, so the whole expression is 1.
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What is the value of \(\sqrt{0.000441}\)?
Answer: B
Count the decimal places: a square root halves them.
\[ 0.000441 = \frac{441}{10^{6}} \]
\[ \sqrt{0.000441} = \frac{\sqrt{441}}{10^{3}} = \frac{21}{1000} = 0.021 \]
Six decimal places become three, which is what separates 0.021 from 0.21.
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1/4 of a number is subtracted from 1/3 of the number and the result is 12. The number is:
Answer: A
Let the number be x.
According to the question, \(\frac{1}{3}x - \frac{1}{4}x = 12\).
To subtract the fractions, find a common denominator, which is 12.
\(\frac{4x - 3x}{12} = 12\).
\(\frac{x}{12} = 12\).
\(x = 144\).
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The sum of three consecutive even numbers is 144. What is the largest of these numbers?
Answer: C
Let the three consecutive even numbers be \(x\), \(x+2\), and \(x+4\).
Their sum is \(x + (x+2) + (x+4) = 144\).
\(3x + 6 = 144\).
\(3x = 138\).
\(x = 46\).
The numbers are 46, 48, and 50. The largest number is 50.
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If you subtract -1 from +1, what is the result?
Answer: C
Subtracting a negative is the same as adding.
\[ 1 - (-1) = 1 + 1 = 2 \]
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What is the approximate value of \(49.98 \times 12.01 \div 2.99\)?
Answer: B
For approximation, we can round the numbers to the nearest whole number.
\(49.98 \approx 50\).
\(12.01 \approx 12\).
\(2.99 \approx 3\).
The expression becomes \(50 \times 12 \div 3\).
Following BODMAS: \(12 \div 3 = 4\).
Then, \(50 \times 4 = 200\).
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The total number of prime numbers between 1 and 50 is:
Answer: C
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The prime numbers between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Counting these numbers, we find there are 15 prime numbers.
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What is the result of \(0.1 \times 0.01 \times 0.001 \times 10^7\)?
Answer: B
First, multiply the decimal numbers.
\(0.1 = 10^{-1}\)
\(0.01 = 10^{-2}\)
\(0.001 = 10^{-3}\)
The expression becomes \(10^{-1} \times 10^{-2} \times 10^{-3} \times 10^7\).
When multiplying powers with the same base, we add the exponents.
\(10^{(-1-2-3+7)} = 10^{-6+7} = 10^1 = 10\).
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A number is as much greater than 36 as is less than 86. Find the number.
Answer: A
Let the number be x.
The condition can be written as an equation: \(x - 36 = 86 - x\).
Add x to both sides: \(2x - 36 = 86\).
Add 36 to both sides: \(2x = 86 + 36 = 122\).
\(x = \frac{122}{2} = 61\).
Alternatively, the number is the average of 36 and 86: \(\frac{36+86}{2} = \frac{122}{2} = 61\).
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Evaluate: \(106 \times 106 - 94 \times 94\)
Answer: A
Use the difference of two squares rather than multiplying out.
\[ 106^{2} - 94^{2} = (106 + 94)(106 - 94) \]
\[ = 200 \times 12 = 2400 \]
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