Math Lair Home > Test Preparation > **Solutions for Practice Test 7, The Official SAT Study Guide, Section 7**

*Practice Test 1*: Sections 3, 7, 8.*Practice Test 2*: Sections 2, 5, 8.*Practice Test 3*: Sections 2, 5, 8.*Practice Test 4*: Sections 3, 6, 9.*Practice Test 5*: Sections 2, 4, 8.*Practice Test 6*: Sections 2, 4, 8.*Practice Test 7*: Sections 3, 7, 9.*Practice Test 8*: Sections 3, 7, 9.*Practice Test 9*: Sections 2, 5, 8.*Practice Test 10*: Sections 2, 5, 8.

Here are solutions for section 7 of practice test #7 in The Official SAT Study Guide, second edition, found on pages 785–790. The solutions below demonstrate faster, more informal methods that might work better for you on a fast-paced test such as the SAT. To learn more about these methods, see my e-book Succeeding in SAT Math or the SAT math tips page.

- 60 is divisible by 3, so 60 − 3 = 57 is divisible by 3. Select (C) 57.
- Try drawing lines through the figures and see if you can find one through which two different lines can be drawn. You will find that the answer is (D) X.
- Try a special case: Say that Bobby does three chores (
`n`= 3). He will make $16 for the week. - Look at the answer choices: See what each answer evaluates to when
`n`= 3:- (A) evaluates to 13
- (B) evaluates to 36
- (C) evaluates to 32
- (D) evaluates to 16
- (E) evaluates to 26

`n`.

- Try a special case: Say that Bobby does three chores (
- Estimate the answer: It doesn't hurt to make a rough estimate of the area of figure
`B`. It appears to definitely be less than the area of figure`A`, but it is more than half as large. Perhaps its area is somewhere around 16–18 sq cm or so. - Look at the answer choices: We can eliminate answer (A), since it is less than half of the area of figure
`A`. The other answers are too close to our estimate for us to be able to eliminate any. - There are 13 little squares in figure
`A`, so each little square has an area of 2 sq cm. There are 8 little squares in figure`B`, so its area must be 2 × 8 = 16. Select (C) 16 sq cm.

- Estimate the answer: It doesn't hurt to make a rough estimate of the area of figure
- Looking at the graph, the only person showing a significant increase (i.e. the shaded bar being much higher than the unshaded bar) is Goldberg. Select (B) Goldberg.
- The average of 6, 6, 12, and 16 is ¼(6 + 6 + 12 + 16) = 10. If
`x`were 10, the average of the 5 numbers would still be 10. Select (D) 10. - Estimate the answer: Compare the angle labelled
`z`° with the angles depicted in the "Reference Information". The angle in question appears to be more than 60°, so maybe 65° or 70° or so. - Look at the answer choices: Since we're fairly certain that the angle must be more than 60°, we can eliminate (A) 55° and (B) 60°.
- Since the angles of a triangle must add to 180°,90 + 2
`y`+ 40 = 180

2`y`= 50`y`= 25 - Again, as the angles of a triangle must add to 180°,90 +Select (C) 65.
`y`+`z`= 180

90 + 25 +`z`= 180`z`= 65

- Estimate the answer: Compare the angle labelled
- If the program selected 13, it would have printed 26, as 13 is odd.
- If the program selected 26, it would have printed 26, as 26 is even.
- If the program selected 52, it would have printed 52, as 52 is even. Since 26 could be printed only if 13 or 26 were selected, select (C) I and II only.

- Try a special case: Say that
`m`= 3 and`s`= 2. There are 60 × 3 + 2 = 182 seconds in 3 minutes and 2 seconds. - Look at the answer choices: See which answer evaluates to 182 when
`m`= 3 and`s`= 2:- (A) evaluates to 182
- (B) evaluates to 123
- (C) evaluates to 300
- (D) evaluates to 1/12
- (E) evaluates to 2 + 1/20

`m`+ 60`s`.

- Try a special case: Say that
- If two expressions multiply to zero, either one expression must be zero, or the other must be zero. So, either 2
`x`− 2 = 0 (Equation 1) or 2 −`x`= 0 (Equation 2). Solving equation 1,`x`= 1. Solving equation 2,`x`= 2. Select (D) 1 and 2 only. - Take the cube root of both sides of the equation:Select (C)
`x`=`y`³`y`³ - Solution 1: A line segment with a slope of −1 would run from the upper left to the lower right, and would be at about a 45° angle to both the
`x`- and`y`-axes. The only such line segment is DC. Select (E) DC. - Solution 2:
- Estimate the answer: A line segment with a slope of −1 will run down as it runs to the right.
- Look at the answer choices:
`OC`and`OD`run up as they run to the right. Their slopes are positive. Eliminate answers (C) and (D). - Draw a diagram: On the diagram, write the values of
`O`(0,0),`D`(1,3),`C`(3,1),`B`(3,−1), and`A`(1,−3). - Guess and check: Find the slopes of lines
`OA`,`OB`and`DC`(slope = (`y`_{2}−`y`_{1})/(`x`_{2}−`x`_{1}). For`DC`, the slope is (1 − 3)/(3 − 1) = −1. Select (E)`DC`.

- Solution 1: A line segment with a slope of −1 would run from the upper left to the lower right, and would be at about a 45° angle to both the
- Look at the answer choices: Look at each answer, and eliminate any answers where any number does not satisfy exactly one criterion:
- In (B), 25 is both odd and a multiple of 5. Eliminate that answer.
- In (C), 15 is both odd and a multiple of 5. Eliminate that answer.
- In (D), 15 is both odd and a multiple of 5. Eliminate that answer.
- In (E), 34 is neither odd, a multiple of 5, nor Kyle's birth date. Eliminate that answer.

- If
`x`> 3, then both sides of the equation are positive. We can then square both sides:Select (C)`x`+ 9 = (`x`− 3)²`x`+ 9 =`x`² − 6`x`+ 9`x`=`x`² − 6`x``x`=`x`² − 6`x`. - To find the number of integers between 1 and 100 that are not squares, we can find the number of squares and subtract that from 100. Now, 1² = 1 and 10² = 100, so there are 10 squares in total between 1 and 100. Subtracting 10 from 100, 100 − 10 - 90. Select (E) 90.
- Draw a diagram: It can help to draw the direct route from
`A`to`D`, as well as the total distance travelled in each direction. See the diagram below, with these lines drawn in thicker: - You can see that the direct distance is the hypotenuse of a right triangle with legs 15 and 20. So, by the Pythagorean theorem, the distance travelled by the direct route is √15² + 20² = 25. The distance from
`A`to`B`to`C`to`D`is 16 + 15 + 4 = 35. So, she would save 35 − 25 = 10 miles by travelling there directly. Select (C) 10.

- Draw a diagram: It can help to draw the direct route from
- The larger circle's area is π(1)² = π. The smaller circle's area is π(½)² = ¼π. The ratio of the area of the larger to the area of the smaller is 1:¼ = 4:1. Select (D) 4:1.
- This question is a lot easier if you realize that the sum of the integers from −22 to 22 is 0. Now, summing the next few integers after 22, we get 23 + 24 + 25 = 72. So, the sum of the integers between −22 and 25 is 72. Select (B) 25.
- If
`x`^{−4/3}=`k`^{−2}, then`x`^{−⅔}=`k`^{−1}= 1/`k` - If
`y`^{4/3}=`n`², then`y`^{−4/3}=`n`^{−2}, and`y`[−⅔] =`n`^{−1}= 1/`n`. - Now that we know
`x`^{−⅔}and`y`^{−⅔}, we can now determine (`xy`)^{−⅔}:(Select (A) 1/(`xy`)^{−⅔}

=`x`^{−⅔}`y`^{−⅔}

=(1/`k`)(1/`n`)

=1/(`nk`)`nk`).

- If
- The second graph is the same as the first graph of
`y`=`f`(`x`), shifted 2 units down and 3 units to the left. So, if the first graph is the graph of`y`=`f`(`x`), the second graph would be the graph of`y`=`f`(`x`− 2) −3. So,`h`= −2 and`k`= −3, and`hk`= 6. Select (E) 6.