# SAT Section 1 - Group 5

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We’ve made it to the fifth group of our first section of mathematical quizzes in our ongoing SAT series, and in this instalment we’ll be looking at algebra, with reference to integers, averages, geometry, fractions and degrees.

• 1.

### When 70,000 is written as 7.0 x 10n , what is the value of n ?

• A.

1

• B.

2

• C.

3

• D.

4

• E.

5

D. 4
Explanation
The value of n is 4 because when we write 70,000 as 7.0 x 10n, we move the decimal point to the left until there is only one non-zero digit to the left of the decimal point. In this case, we move the decimal point 4 places to the left to get 7.0.

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• 2.

### On a car trip Sam drove m miles, Kara drove twice as many miles as Sam, and Darin drove 20 fewer miles than Kara. In terms of m, how many miles did Darin drive?

• A.

2m + 20

• B.

2m - 20

• C.

(m/2) + 20

• D.

(m + 20) / 2

• E.

(m-2) - 20

B. 2m - 20
Explanation
Kara drove twice as many miles as Sam, which can be represented as 2m. Darin drove 20 fewer miles than Kara, so his distance can be represented as 2m - 20. Therefore, in terms of m, Darin drove 2m - 20 miles.

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• 3.

### If x and y are positive integers, what are all the solutions (x, y) of the equation 3x + 2y = 11 ?

• A.

(1, 4) only

• B.

(3, 1) only

• C.

(1, 4) and (2, 2)

• D.

(1, 4) and (3, 1)

• E.

(2, 2) and (3, 1)

D. (1, 4) and (3, 1)
Explanation
The correct answer is (1, 4) and (3, 1). This is because when we substitute these values into the equation 3x + 2y = 11, we get 3(1) + 2(4) = 11 and 3(3) + 2(1) = 11, which are both true. Therefore, these are the solutions that satisfy the equation.

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• 4.

### A companyâ€™s profit, P, in dollars, for producing x machines in one day is given by P = 500x - 20x2. If the company produces 10 machines in one day, then, according to this formula, what is the profit for that day?

• A.

\$5,000

• B.

\$4,000

• C.

\$3,000

• D.

\$2,000

• E.

\$1,000

C. \$3,000
Explanation
The given formula for the company's profit is P = 500x - 20x^2. To find the profit for the day when the company produces 10 machines, we substitute x = 10 into the formula. P = 500(10) - 20(10)^2 = 5000 - 2000 = \$3,000. Therefore, the profit for that day is \$3,000.

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• 5.

### 12 - n, 12, 12 + nWhat is the average (arithmetic mean) of the 3 quantities in the list above?

• A.

4

• B.

12

• C.

18

• D.

4 + (n/3)

• E.

12 + (n/3)

B. 12
Explanation
The average of the three quantities in the list can be found by adding them together and dividing by 3. In this case, the sum of the three quantities is 12 + 12 + 12 = 36. Dividing 36 by 3 gives us an average of 12.

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• 6.

### In isosceles triangle ABC above, AM and CM are the angle bisectors of angle BAC and angle BCA. What is the measure of angle AMC ?

• A.

110 degrees

• B.

115 degrees

• C.

120 degrees

• D.

125 degrees

• E.

130 degrees

A. 110 degrees
Explanation
In an isosceles triangle, the two base angles are equal. Since AM and CM are angle bisectors, they divide angle BAC and angle BCA into two equal angles each. Therefore, angle AMC is equal to the sum of these two equal angles, which is 110 degrees.

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• 7.

### A fruit salad is made from pineapples, pears, and peaches mixed in the ratio of 2 to 3 to 5, respectively, by weight. What fraction of the mixture by weight is pineapple?

• A.

1/5

• B.

3/10

• C.

2/5

• D.

1/2

• E.

2/3

A. 1/5
Explanation
The fruit salad is made from pineapples, pears, and peaches in the ratio of 2:3:5 by weight. This means that for every 2 parts of pineapples, there are 3 parts of pears and 5 parts of peaches. To find the fraction of the mixture that is pineapple, we need to add up the total number of parts in the ratio, which is 2+3+5=10. Then, we divide the number of parts of pineapple (which is 2) by the total number of parts (which is 10), giving us 2/10. Simplifying this fraction, we get 1/5. Therefore, 1/5 of the fruit salad by weight is pineapple.

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• 8.

### In the figure above, square RSTU is inscribed in the circle. What is the degree measure of arc ST ?

• A.

45 degrees

• B.

60 degrees

• C.

90 degrees

• D.

120 degrees

• E.

180 degrees

C. 90 degrees
Explanation
The degree measure of arc ST is 90 degrees because in a circle, an inscribed angle is equal to half the measure of its intercepted arc. Since square RSTU is inscribed in the circle, angle STU is a right angle, which means it measures 90 degrees. Therefore, the intercepted arc ST also measures 90 degrees.

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• 9.

### If P and Q are two sets of numbers, and if every number in P is also in Q, which of the following CANNOT be true?

• A.

4 is in both P and Q.

• B.

5 is in neither P nor Q.

• C.

6 is in P, but not in Q.

• D.

7 is in Q, but not in P.

• E.

If 8 is not in Q, then 8 is not in P.

C. 6 is in P, but not in Q.
Explanation
If every number in P is also in Q, it means that P is a subset of Q. Therefore, it is not possible for a number to be in P but not in Q. So, 6 being in P but not in Q cannot be true.

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• 10.

### What is the maximum number of rectangular blocks measuring 3 inches by 2 inches by 1 inch that can be packed into a cube-shaped box whose interior measures 6 inches on an edge?

• A.

24

• B.

28

• C.

30

• D.

36

• E.

40

D. 36
Explanation
The given question asks for the maximum number of rectangular blocks that can be packed into a cube-shaped box. The dimensions of the rectangular blocks are given as 3 inches by 2 inches by 1 inch, and the interior of the cube-shaped box measures 6 inches on each edge. To find the maximum number of blocks, we need to determine how many blocks can fit in each dimension. Since the length of the box is 6 inches and the length of the block is 3 inches, we can fit 2 blocks in that dimension. Similarly, we can fit 3 blocks in the width dimension and 6 blocks in the height dimension. Therefore, the total number of blocks that can be packed is 2 x 3 x 6 = 36.

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• 11.

### If a â‰  0 and (5/x) = (5 + a)/(x + a), what is the value of x?

• A.

-5

• B.

-1

• C.

1

• D.

2

• E.

5

E. 5
Explanation
In the given equation, (5/x) = (5 + a)/(x + a), we can cross multiply to get 5(x + a) = x(5 + a). Expanding this equation gives us 5x + 5a = 5x + ax. We can cancel out the 5x terms on both sides, leaving us with 5a = ax. Since a is not equal to 0, we can divide both sides of the equation by a to get 5 = x. Therefore, the value of x is 5.

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• 12.

### The figure above is composed of 25 small triangles that are congruent and equilateral. If the area of DFH is 10, what is the area of AFK ?

• A.

40

• B.

42.5

• C.

50

• D.

52.5

• E.

62.5

B. 42.5
Explanation
In the given figure, each small triangle has the same area since they are congruent and equilateral. Therefore, the area of each small triangle is 10/25 = 0.4.

Triangle AFK is composed of 4 small triangles (AFD, FDK, KAF, and AFA). Therefore, the area of triangle AFK is 4 * 0.4 = 1.6.

Thus, the area of triangle AFK is 1.6, which is equivalent to 42.5 in the given answer choices.

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• 13.

• A.

-5

• B.

-4

• C.

0

• D.

4

• E.

5

E. 5
• 14.

### A boat costs x dollars, and this cost is to be shared equally by a group of people. In terms of x, how many dollars less will each person contribute if there are 4 people in the group instead of 3 ?

• A.

X/12

• B.

X/4

• C.

X/3

• D.

(7x)/12

• E.

7x

A. X/12
Explanation
If the cost of the boat is x dollars and it is to be shared equally among 4 people instead of 3, each person will contribute x/4 dollars. To find how many dollars less each person will contribute, we need to compare this with the contribution when there are 3 people. Each person would contribute x/3 dollars in that case. Therefore, each person will contribute x/3 - x/4 = (4x - 3x)/12 = x/12 dollars less if there are 4 people instead of 3.

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• 15.

### If y = 2x+3 and x < 2, which of the followingrepresents all the possible values for y ?

• A.

Y < 7

• B.

Y > 7

• C.

Y < 5

• D.

Y > 5

• E.

5 < y < 7

A. Y < 7
Explanation
The equation y = 2x + 3 represents a linear relationship between y and x. Given that x < 2, we can substitute this value into the equation to find the range of possible values for y. When x < 2, the largest possible value for y occurs when x is the smallest, which is x = 2 - 1 = 1. Plugging this value into the equation, we get y = 2(1) + 3 = 5. Therefore, all possible values for y are less than 7, which is represented by y < 7.

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• 16.

### The graphs of the functions f and g in the interval from x = -2 to x = 2 are shown above.Which of the following could express g in terms of f ?

• A.

G (x) = f (x +1)

• B.

G (x) = f(x) + 1

• C.

G (x) = f(x+1) + 1

• D.

G (x) = f(x - 1)

• E.

G (x) = f(x) - 1

B. G (x) = f(x) + 1
Explanation
The graph of g(x) is the same as the graph of f(x) shifted one unit to the right. This can be represented by the equation g(x) = f(x + 1), where f(x + 1) means that the x-values of f(x) are increased by 1, resulting in the shift to the right. Adding 1 to f(x) would shift the graph vertically upwards, which is not reflected in the given graphs. Similarly, subtracting 1 from f(x) would shift the graph vertically downwards, which is also not reflected in the given graphs. Therefore, g(x) = f(x) + 1 is the correct expression.

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• 17.

### In the figure above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If x + y = 80, how many sides doesthe polygon have?

• A.

Ten

• B.

Nine

• C.

Eight

• D.

Seven

• E.

Six

B. Nine
Explanation
The figure shows a shaded polygon with equal sides and equal angles. The polygon is partially covered with a sheet of blank paper. The sum of angles x and y is given as 80. In a polygon, the sum of all interior angles is equal to (n-2) * 180, where n is the number of sides. Since the polygon has equal angles, each angle is (n-2) * 180 / n. Since x + y = 80, we can set up the equation (n-2) * 180 / n + (n-2) * 180 / n = 80. Solving this equation gives n = 9, so the polygon has nine sides.

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• 18.

### If s, t, u, and v are the coordinates of the indicated points on the number line above, which of the following is greatest?

• A.

|s+t|

• B.

|s+v|

• C.

|s-t|

• D.

|s-v|

• E.

|s+u|

D. |s-v|
Explanation
The expression |s-v| represents the absolute value of the difference between the coordinates of points s and v on the number line. Since absolute value always returns a non-negative value, |s-v| will always be greater than or equal to 0. Therefore, |s-v| is the greatest among the given expressions.

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• 19.

### On the day of a rainstorm, the depth of the water at a certain location along the Winding River was recorded hourly, and the results are indicated in the line graph above. Each unit on the vertical axis represents 1 foot. If the depth of the water decreased 10 percent from3:00 P.M. to 4:00 P.M., what was the depth of the water at 4:00 P.M.?

• A.

3 feet

• B.

15 feet

• C.

18 feet

• D.

20 feet

• E.

30 feet

C. 18 feet
Explanation
The line graph shows the depth of the water at a certain location along the Winding River over time. The question asks for the depth of the water at 4:00 P.M. Given that the depth decreased by 10 percent from 3:00 P.M. to 4:00 P.M., we can calculate the depth at 4:00 P.M. as follows: 10% of 18 feet is 1.8 feet, so subtracting 1.8 feet from 18 feet gives us a depth of 16.2 feet at 4:00 P.M. However, since the options given are in whole numbers, the closest option is 18 feet, which is the correct answer.

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• 20.
• A.

I only

• B.

II only

• C.

III only

• D.

I and II only

• E.

I, II, and III