A number increased by 22.5% gives 98. The number is:
Answer: C
Let the number be x.
When the number is increased by 22.5%, it becomes 100% + 22.5% = 122.5% of itself.
So, \(1.225 \times x = 98\).
\(x = \frac{98}{1.225} = \frac{98000}{1225}\).
To simplify, divide both by 25: \(\frac{3920}{49}\).
Now divide by 49: \(3920 / 49 = 80\).
So, the number is 80.
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What is the value of \((81)^{-1/4}\) ?
Answer: B
A negative exponent inverts and a fractional one takes a root.
\[ 81^{-1/4} = \frac{1}{81^{1/4}} = \frac{1}{\sqrt[4]{81}} \]
\[ = \frac{1}{3} \]
Since \( 3^{4} = 81 \). Missing the minus sign would give 3.
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What is 0.09% of 25% of 1200?
Answer: A
Work inward to outward.
\[ 25\% \text{ of } 1200 = 300 \]
\[ 0.09\% \text{ of } 300 = 0.0009 \times 300 = 0.27 \]
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If the sum of five consecutive integers is 150, what is the middle integer?
Answer: C
For any set of consecutive integers, the sum is the number of integers multiplied by their average. The average is also the middle number.
Average = Sum / Number of integers = 150 / 5 = 30.
So, the middle integer is 30. The integers are 28, 29, 30, 31, 32.
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Simplify: \( (\frac{1}{2})^{-2} + (\frac{1}{3})^{-2} + (\frac{1}{4})^{-2} \)
Answer: A
A negative exponent inverts the base. \((a/b)^{-n} = (b/a)^n\).
\((\frac{1}{2})^{-2} = (2)^2 = 4\).
\((\frac{1}{3})^{-2} = (3)^2 = 9\).
\((\frac{1}{4})^{-2} = (4)^2 = 16\).
The sum is \(4 + 9 + 16 = 29\).
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Which of the following is an irrational number?
Answer: C
An irrational number cannot be expressed as a simple fraction and its decimal representation is non-terminating and non-repeating.
A) \(\sqrt{16} = 4\), which is rational.
B) \(\sqrt{(144/9)} = \sqrt{16} = 4\), which is rational.
C) \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\). Since \(\sqrt{3}\) is irrational, \(2\sqrt{3}\) is also irrational.
D) \(\sqrt{0.04} = \sqrt{4/100} = 2/10 = 0.2\), which is rational.
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What is the place value of 5 in the number 67.058?
Answer: B
In 67.058 the digits after the point run tenths, hundredths, thousandths:
\[ 0,\ 5,\ 8 \]
The 5 occupies the second place, so it is 5 hundredths, or 0.05.
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In a class, there are 30 boys and 20 girls. What fraction of the students are boys?
Answer: C
Total number of students = Number of boys + Number of girls = 30 + 20 = 50.
Number of boys = 30.
Fraction of boys = \(\frac{\text{Number of boys}}{\text{Total students}} = \frac{30}{50}\).
Simplifying the fraction by dividing the numerator and denominator by 10 gives \(\frac{3}{5}\).
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Simplify: \(1.25 \times 40 - 2.5 \times 10\)
Answer: A
\[ 1.25 \times 40 = 50, \qquad 2.5 \times 10 = 25 \]
\[ 50 - 25 = 25 \]
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A number exceeds its two-fifth by 75. The number is:
Answer: A
Let the number be x.
Two-fifth of the number is \(\frac{2}{5}x\).
The equation is \(x = \frac{2}{5}x + 75\).
\(x - \frac{2}{5}x = 75\).
\(\frac{3}{5}x = 75\).
\(x = 75 \times \frac{5}{3} = 25 \times 5 = 125\).
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