Carlos and Katherine are estimating acceleration by rolling a ball from rest down a ramp. At 1 second, the ball is moving at 5 meters per second (m/s); at 2 seconds, the ball is moving at 10 m/s; at 3 seconds, the ball is moving at 15 m/s; and at 4 seconds, it is moving at 20 m/s. When graphed on an xy-plane, which equation best describes the ball’s estimated acceleration where y expresses speed and x expresses time?
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Solution
Figure out the points that will be on the graph from the data given: (0, 0), (1, 5), (2, 10), (3, 15), (4, 20).
Draw a line through or close to these points to get an idea of what the graph will look like. Then use POE.
The line is linear, not quadratic, so you can eliminate (D).
It is also clear that the line begins at the origin, so the y-intercept will be 0.
This will eliminate (A).
A slope of 25 is far too big-ballpark-so you can eliminate (B), leaving (C).
If f(x) = 2x2 + 4 for all real numbers x, which of the following is equal to f(3) + f(5) ?
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Solution
To find the value of f(3) + f(5), find the values of f(3) and f(5) separately: f(3) = 2(3)2 + 4 = 22 and f(5) = 2(5)2 + 4 = 54.
So f(3) + f(5) = 76.
You can tell that f(4) will be between 22 and 54, so you can cross out (A).
If you ballpark (C) and (D), putting 10 or 15 in the function will give you a number bigger than 100, and you're looking for 76, so (C) and (D) are too big.
That means the answer is (B) by POE.
Line l contains points (3, 2) and (4, 5). If line m is perpendicular to line l, then which of the following could be the equation of line m ?
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Solution
First, find the slope of line l by using the slope formula:
\(\frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{5 - 2}{4 - 3} = \frac{3}{1}\)
A line perpendicular to line l must have a slope that is the negative reciprocal of l's slope. So, its slope should be -1⁄3.
In the standard form of a line Ax + By = C, the slope is -A⁄B.
Only (B) has a slope of -1⁄3.
If you didn't remember the rule about the slope of perpendicular lines in standard form, you could have converted the answers to slope-intercept form and sketched out each of the lines to look for the answer that looked perpendicular to l.
x y
-3 -7
-1 -3
2 3
Based on the chart above, which of the following could express the relationship between x and y ?
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Solution
Plug in the values from the chart! Use the pair (-3, -7) from the top of the chart and eliminate answers that are not true: (A) becomes -7 = -3 - 4, which is true. Keep it.
Keep (B): -7 = 2(-3) - 1 is true.
Get rid of (C), which becomes -7 = 2(-3) + 2: -7 does not equal -4.
Get rid of (D): -7 = 3(-3) - 3. -7 does not equal -12.
Now use another pair just to test (A) and (B).
Using (-1, -3), (A) gives -3 = -1 - 4, which is not true, so eliminate it, leaving only (B).
The values (-1, -3) work: -3 = 2(-1) - 1.
If f (x) = \(\sqrt{3x – 2}\), what is the smallest possible value of f (x) ?
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Solution
On this question you can use Plugging In the Answers.
The numbers in the answer choices replace the f(x) portion of the equation, so you can just write out the rest of it, \(\sqrt{3x - 2}\), next to each to see if it can be true.
Start with (A) since you are looking for the smallest value of f(x).
If 0 = \(\sqrt{3x - 2}\), then 0 = 3x - 2 when you square both sides.
Add 2 to both sides to get 2 = 3x, and then divide both sides by 3.
You get x = 2⁄3.
Since this is a real value, the equation works, so the smallest value of f(x) is 0.
Choice (A) is correct.
Of all the houses in a certain neighborhood, 80% have garages. Of those houses with garages, 60% have two-car garages. If there are 56 houses with garages that are not two-car garages, how many houses are there in the neighborhood?
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Solution
Start by figuring out what percent of the houses do not have two-car garages.
Since 60% of the houses with garages have two-car garages, 40% of the houses with garages do not have two-car garages.
In other words, 40% of 80% of the houses do not have two-car garages.
Translate that into math to get \(\frac{40}{100} \times \frac{80}{100}\) = 0.32 or 32% of the houses.
The problem tells us that 56 houses do not have two-car garages, which means 32% of the houses equals 56.
Translating into math gives us \(\frac{32}{100} \times x = 56\).
Solve for x, and you'll get 175, which is (D).
| Milligrams of Gold | |||||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | |
| Limestone | 0.45 | 0.58 | 0.55 | 0.42 | 0.41 |
| Granite | 0.94 | 0.87 | 0.82 | 0.55 | 0.73 |
| Gneiss | 0.38 | 0.60 | 0.37 | 0.40 | 0.34 |
Five samples of each of three different rock types were collected on a hiking trip in Colorado. Each sample was analyzed for its gold content. The milligrams of gold found in each sample are presented in the table above. What is the percent difference of the average gold content in the granite samples when compared to the average gold content of the gneiss samples?
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Solution
The average gold content in the granite samples can be calculated as follows:
\(\frac{0.94 + 0.87 + 0.82 + 0.55 + 0.73}{5} = 0.782\)
The average gold content in the gneiss samples can be calculated as
\(\frac{0.38 + 0.60 + 0.37 + 0.40 + 0.34}{5} = 0.418\)
Because the average gold content in the granite samples is higher, you can eliminate (A) and (C).
Ballpark to find the right answer.
0.782 is almost twice as much as 0.418.
Therefore, granite contains, on average, almost 100% more gold than gneiss does.
The correct answer is (D).
| SPICE PRICES OF DISTRIBUTOR D | |
| Spice | Price Per Pound |
| Cinnamon | $8.00 |
| Nutmeg | $9.00 |
| Ginger | $7.00 |
| Cloves | $10.00 |
The owner of a spice store buys 3 pounds each of cinnamon, nutmeg, ginger, and cloves from distributor D. She then sells all of the spices at $2.00 per ounce. What is her total profit, in dollars?
(1 pound = 16 ounces)
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Solution
This is a hard question, so you have to stay on your toes.
If the owner buys 3 pounds of each spice, that means she pays the following amounts for each spice:
cinnamon: $8 × 3 = $24
nutmeg: $9 × 3 = $27
ginger: $7 × 3 = $21
cloves: $10 × 3 = $30
So she pays a total of 24 + 27 + 21 + 30, or $102 for 12 pounds of spices.
She then sells the spices per ounce, so you have to figure out first how many ounces of spices she has.
If 1 pound is 16 ounces, then 12 pounds is 12 × 16, or 192 ounces.
She sells all the spices at $2 per ounce, so she makes 192 × $2, or $384.
To figure out her profit, subtract the amount she paid for the spices from the amount she made selling them: $384 - $102 = $282, (B).
A total of 140,000 votes were cast for two candidates, Skinner and Whitehouse. If Skinner won by a ratio of 4 to 3, how many votes were cast for Whitehouse?
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Solution
Since this is a ratio question, let's draw a Ratio Box.
We know the ratio for the votes for Skinner and Whitehouse, and the total number of votes cast.
Fill in the total by adding the ratio (4 + 3 = 7), and then find the multiplier by seeing how many times 7 goes into 140,000 (140,000 ÷ 7 = 20,000).
Skinner Whitehouse Total 4 3 7 × 20,000 × 20,000 × 20,000 80,000 60,000 140,000 The question wants to know how many votes Whitehouse received, which is 60,000, (C).
The amount of time that Amy walks is directly proportional to the distance that she walks. If she walks a distance of 2.5 miles in 50 minutes, how many miles will she walk in 2 hours?
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Solution
Since we know the time that Amy walked and the distance she walked are directly proportional, we can set up a proportion to show her distance ÷ time.
We want to know how many miles she'll walk in two hours, so put 120 (60 × 2) minutes in the second half of the ratio: \( \frac{2.5}{50} = \frac{x}{120}\).
To solve, cross-multiply, and you'll get 50x = 2.5 × 120; 50x = 300; x = 6 miles, which is (C).