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|---|---|---|
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A circular logo is enlarged to fit the lid of a jar. The new diameter is 60% larger than the original. By what percentage has the area of the logo increased?
| 32\(\frac{1}{2}\)% | |
| 27\(\frac{1}{2}\)% | |
| 22\(\frac{1}{2}\)% | |
| 30% |
The area of a circle is given by the formula A = πr2 where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))2. If the diameter of the logo increases by 60% the radius (and, consequently, the total area) increases by \( \frac{60\text{%}}{2} \) = 30%
If \( \left|a - 2\right| \) - 4 = 1, which of these is a possible value for a?
| -11 | |
| -3 | |
| -2 | |
| -12 |
First, solve for \( \left|a - 2\right| \):
\( \left|a - 2\right| \) - 4 = 1
\( \left|a - 2\right| \) = 1 + 4
\( \left|a - 2\right| \) = 5
The value inside the absolute value brackets can be either positive or negative so (a - 2) must equal + 5 or -5 for \( \left|a - 2\right| \) to equal 5:
| a - 2 = 5 a = 5 + 2 a = 7 | a - 2 = -5 a = -5 + 2 a = -3 |
So, a = -3 or a = 7.
On average, the center for a basketball team hits 30% of his shots while a guard on the same team hits 40% of his shots. If the guard takes 30 shots during a game, how many shots will the center have to take to score as many points as the guard assuming each shot is worth the same number of points?
| 24 | |
| 40 | |
| 27 | |
| 23 |
guard shots made = shots taken x \( \frac{\text{% made}}{100} \) = 30 x \( \frac{40}{100} \) = \( \frac{40 x 30}{100} \) = \( \frac{1200}{100} \) = 12 shots
The center makes 30% of his shots so he'll have to take:
shots made = shots taken x \( \frac{\text{% made}}{100} \)
shots taken = \( \frac{\text{shots taken}}{\frac{\text{% made}}{100}} \)
to make as many shots as the guard. Plugging in values for the center gives us:
center shots taken = \( \frac{12}{\frac{30}{100}} \) = 12 x \( \frac{100}{30} \) = \( \frac{12 x 100}{30} \) = \( \frac{1200}{30} \) = 40 shots
to make the same number of shots as the guard and thus score the same number of points.
What is \( \frac{8}{8} \) - \( \frac{2}{16} \)?
| \( \frac{1}{16} \) | |
| \(\frac{7}{8}\) | |
| 2 \( \frac{7}{16} \) | |
| 1 \( \frac{7}{16} \) |
To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 16 are [16, 32, 48, 64, 80, 96]. The first few multiples they share are [16, 32, 48, 64, 80] making 16 the smallest multiple 8 and 16 share.
Next, convert the fractions so each denominator equals the lowest common multiple:
\( \frac{8 x 2}{8 x 2} \) - \( \frac{2 x 1}{16 x 1} \)
\( \frac{16}{16} \) - \( \frac{2}{16} \)
Now, because the fractions share a common denominator, you can subtract them:
\( \frac{16 - 2}{16} \) = \( \frac{14}{16} \) = \(\frac{7}{8}\)
What is \( \frac{10\sqrt{28}}{2\sqrt{4}} \)?
| \(\frac{1}{5}\) \( \sqrt{\frac{1}{7}} \) | |
| 7 \( \sqrt{5} \) | |
| 5 \( \sqrt{7} \) | |
| \(\frac{1}{5}\) \( \sqrt{7} \) |
To divide terms with radicals, divide the coefficients and radicands separately:
\( \frac{10\sqrt{28}}{2\sqrt{4}} \)
\( \frac{10}{2} \) \( \sqrt{\frac{28}{4}} \)
5 \( \sqrt{7} \)