2000 CEMC Gauss (Grade 8) Problems/Problem 9

Problem

Of the following five statements, how many are correct?

$\text{(i) } 20\% \text{ of } 40 = 8 \qquad \text{(ii) } 2^{3} = 8 \qquad \text{(iii) } 7 - 3 \times 2 = 8 \qquad \text{(iv) } 3^2 - 1^2 = 8 \qquad \text{(v) } 2(6 - 4)^2 = 8$

$\text{ (A) }\ 1 \qquad\text{ (B) }\ 2 \qquad\text{ (C) }\ 3 \qquad\text{ (D) }\ 4 \qquad\text{ (E) }\ 5$

Solution 1

We can just simply verify whether or not each statement is true, and count how many are true:

$20\% \times 40 = \frac{1}{5} \times 40 = 8$ , so the first statement is true

$2^{3} = 8$ , so the second statement is true

$7 - 3 \times 2 = 7 - 6 = 1$ , so the third statement is false

$3^{2} - 1^2 = 9 - 1 = 8$ , so the fourth statement is true

$2(6 - 4)^2 = 2(2)^2 = 2 \times 4 = 8$ , so the last statement is true

Four of these are correct. Thus, the answer is $\boxed {\textbf {(D) } 4}$ .

~anabel.disher

Solution 1.5

We can also use the fact that $a^{2} - b^{2} = (a - b)(a + b)$ , or difference of squares, to verify $3^{2} - 1^{2}$ :

$3^{2} - 1^{2} = (3 - 1)(3 + 1) = 4 \times 2 = 8$

The answer is $\boxed {\textbf {(D) } 4}$ .

~anabel.disher

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2000 CEMC Gauss (Grade 8) Problems/Problem 9

Problem

Solution 1

Solution 1.5