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## Question Explanations For

##
*Sample SAT Math Test (Calculator Section)*

## 1.

The Correct Answer is (B) —
Because the scientist took a random sample, we can assume that all other squares have approximately 15 cells. To get the total number of cells, we can multiply the number of cells per square by the total number of squares, giving us 15n.

## 2.

The Correct Answer is (C) —
The equation y=-2x+1 is in the form y=ax+b, meaning that the slope is -2 and the y-intercept is 1. Therefore, we expect the graph to have a negative slope and a positive y-intercept. Answer choice (C) is the only graph that satisfies these criteria.

## 3.

The Correct Answer is (A) —
When we plug in 10 for x in f(x), we get 1000/(10-20)=1000/(-10)=-100.

## 4.

The Correct Answer is (A) —
When we plug in 10 for y in y = 0.75x + 10, we get 10 = 0.75x + 10. Subtracting 10 from each side gives us 0 = 0.75x. Dividing each side by 0.75 gives us x = 0.

## 5.

The Correct Answer is (C) —
The probability that the first marble drawn will be blue is 6/13. Since we do not replace this marble, the probability that the second marble drawn will be blue is 5/12. Multiplying 6/13 and 5/12 gives us 5/26.

## 6.

The Correct Answer is (D) —
When we plug in the second equation for y in the first equation, we get 10x - 2(5x - 4) + 3 = 11. Distributing the 2 into the parentheses gives us 10x - 10x + 8 + 3 = 11, which simplifies to 11 = 11. Since this statement is always true, there are infinite solutions to the system of equations.

## 7.

The Correct Answer is (B) —
Taking the square root of both sides of the equation gives us x + 20 = 70 - x. Adding x and subtracting 20 from both sides gives us 2x = 50. Dividing each side by 2 gives us x = 25.

## 8.

The Correct Answer is (A) —
Average speed is calculated by dividing the total distance traveled by the total time. Driver 1 traveled 300 miles in 10 hours so his average speed is 30 miles per hour. Driver 2 traveled 200 miles in 10 hours so his average speed is 20 miles per hour. The difference in the two average speeds is 10 miles per hour.

## 11.

The Correct Answer is (B) —
Solving the second equation for y, we get y = 34 - x. Plugging this value for y into the first equation, we get 1.5x + 2(34 - x) = 58. Distributing the 2 into the parentheses gives us 1.5x + 68 - 2x = 58. Combining the x terms and subtracting 68 from each side gives us -0.5x = -10. Dividing each side by -0.5 gives us x = 20.

## 12.

The Correct Answer is (D) —
Since 10 out of the 100 butterflies caught were tagged, we can assume that 10% of the entire butterfly population has been tagged. This would mean that 50 (the original number of tagged butterflies) is 10% of the entire population, and 500 would be a good estimate for the total number of butterflies.

## 17.

The Correct Answer is (A) —
The ratio for the solution is 10 mg/80 uL. When we equate this to the ratio that we want to solve, we get 10 mg/80 uL = 300 mg/x . Solving for x we get x = 2400 uL. Converting this to liters, we get 2400 uL(L/1,000,000uL) = 0.0024 L which converts to 2.4 mL.

## 18.

The Correct Answer is (D) —
The remaining distance can be calculated by subtracting the total distance traveled from 600 miles. The distance traveled is speed multiplied by time, which in this case is 50t. Subtracting this from 600 gives us f(t) = 600 - 500t.

## 19.

The Correct Answer is (D) —
Plugging in -1 for x in the inequality gives us 4y > -8 which simplifies to y > -2. This eliminates answer choices (A) and (B). Plugging in choice (C) gives us 4(-2) > 3(1) - 5 which simplifies to -8> -2 which is an untrue statement. This leaves us with answer choice (D).

## 20.

The Correct Answer is (D) —
Since Stacey can make 42 cookies per one gallon of milk, we can find the number of cookies she can bake by multiplying the number of gallons by 42. To find the number of gallons of milk she can buy with $10, we divide $10 by $2.40. We then multiply this value by 42 giving us $10/$2.40 per gallon x 42 cookies/gallon = 175 cookies.

## 21.

The Correct Answer is (D) —
Solving for f(23) we get f(23) = 2(23) - 10 = 46 - 10 = 36. Plugging in this value for g(x) gives us g(36) = 36/3 + 1 = 12 + 1 = 13.

## 22.

The Correct Answer is (C) —
We first need to calculate what percent of the sample is in favor of the bridge. Dividing the total number in favor by the total number of people surveyed, we get
32/62 = 0.516. We can assume that this same fraction of the total population will also be in favor of the bridge. Multiplying this fraction by 700 gives us (0.516)(700) = 361.2 which we can round to 361.

## 23.

The Correct Answer is (C) —
If we recognize that the first equation can be rewritten as a difference of squares, we get (x + y)(x - y) = 9. Substituting the second equation for x - y , we get (x + y)(1) = 9 , which gives us x + y = 9.

## 24.

The Correct Answer is (C) —
When we set f(x) equal to zero, we can see that (x + 2), (x - 2), and (x + a) and can be used to find the roots of the function, giving us x-intercepts of x = -2, x =2, and x = -a. We see that the function crosses the x-axis at what looks like -2, -1, and 2, meaning that -1 = -a which gives us a = 1.

## 27.

The Correct Answer is (C) —
The standard deviation is a measure of variance calculated by considering how far away each data point is away from the mean, so answer choice (C) is correct. Having fewer data points does not necessarily mean a smaller standard deviation, so answer choice (A) is incorrect. We do not have information to determine whether the difference in standard deviation is statistically significant, so answer choice (B) is incorrect. We are also given no information on the range, so answer choice (D) is incorrect.

## 29.

The Correct Answer is (B) —
Since GDP per capita is calculated by dividing the GDP by the total number of people, we can calculate the total number of people by dividing the GDP by the GDP per capita. This gives us $135,000,000/$30,000 = 4500.

## 30.

The Correct Answer is (D) —
Using the same method as problem 29, we find that the population in 1990 was $120,000,000/$32,000 = 3750 and that the population in 2010 was $147,000,000/$28,000 = 5250. Dividing the population in 2010 by the population in 1990, we get 5250/3750 = 1.40, meaning that the population in 2010 was 40% greater than the population in 1990.

## 31.

The Correct Answer is (0<y<4) —
Plugging in x =2 into the inequality gives us y + 6 < 10 and subtracting 6 from each side gives us y < 4. The problem asks only for positive values so any value of y greater than 0 but less than 4 is a valid answer.

## 32.

The Correct Answer is (100) —
If m is the number of minutes, the cost of Plan A can be written as 20 + 0.40m. Similarly, the cost of Plan B can be written as 10 + 0.50m. Setting these two expressions equal to one another, we get 20 + 0.40m = 10 + 0.50m. Combining like terms gives us 10 = 0.10m. Solving for m gives us m = 100 minutes.

## 33.

The Correct Answer is (60) —
Multiplying each side of the equation by 3 gives us p - 15 = 3q/4. Then multiplying each side by 4 gives us 4p - 60 = 3q. Moving the -60 to the right and the 3q to the left gives us 4p - 3q = 60.

## 34.

The Correct Answer is (12) —
This is a counting problem. First place can be either Carlos or Diana, so there are two possibilities for first place. Once the first place person is chosen, there is only one option left (either Carlos or Diana) for second place. Third place could be one of the three remaining people and fourth place could one of the remaining two people, leaving only one option for fifth place. This gives us 2 x 1 x 3 x 2 x 1 = 12.

## 35.

The Correct Answer is (6/5 or 1.2) —
Simplifying the numerators on both sides of the equation gives us (12 - m)/4 = (3m + 18)/8. Cross-multiplying gives us 96 - 8m = 12m + 72. Adding 8m and subtracting 72 from each side gives us 24 = 20m. Dividing each side by 20 gives us m = 24/20 = 6/5.

## 36.

The Correct Answer is (200) —
If x is the width, then 2x is the length of the pasture. The perimeter of the pasture is x + x + 2x + 2x = 6x. Since fencing costs $4 per meter, and the farmer spent $2400, the fence must have been $2400/$4 per meter = 600 meters. Setting this equal to 6x gives us x = 100 meters. The pasture’s longest side = 2x = 200 meters.

## 37.

The Correct Answer is (9600) —
The first $30,000 of Liu’s income has a tax rate of 10% so the tax is $30,000(0.1) = $3000. The tax rate on the next $40,000 is 14% so the tax is $40,000(0.14) = $5600. The last $5000 has a tax rate of 20%, making the tax $1000. Adding all the taxes up gives us $3000 + $5600 + $1000 = $9600.

## 38.

The Correct Answer is (600) —
At a flat rate of 14%, the tax on $80,000 is $11,200. In the progressive system, The first $30,000 of Kristof’s income has a tax rate of 10% so the tax is $30,000(0.1) = $3000. The tax rate on the next $40,000 is 14% so the tax is $40,000(0.14) = $5600. The last $10,000 has a tax rate of 20%, making the tax $2000. Adding all the taxes up gives us $3000 + $5600 + $2000 = $10,600. The difference between $11,200 and $10,600 is $600.