Math Lair Home > Test Preparation > **Solutions for Practice Test 8, 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 #8 in The Official SAT Study Guide, second edition, found on pages 847–852. 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.

- The sum of the letters assigned to the word "exquisite" is 1 + 5 + 5 + 1 + 1 + 1 + 1 + 1 + 1 = 17. Select (B) 17.
- Divide both sides of the given equation by 2:Select (B) 10.
`x`− 5 = 10 - Try a special case: Say that
`t`= 1. - If
`t`= 1, then (A) evaluates to 3, (B) evaluates to 1, (C) evaluates to 1, (D) evaluates to 5, and (E) evaluates to 6. Select (E) 5`t`+ 1.

- Try a special case: Say that
- Estimate the answer: The length of each side of the equilateral triangle is definitely smaller than the length of
`DF`(9), but significantly larger than the length of`ED`(4). Perhaps the length is 6 or 7. - Look at the answer choices: The only answer given that is significantly larger than 4 but less than 9 is (C) 7. Select that answer.

- Estimate the answer: The length of each side of the equilateral triangle is definitely smaller than the length of
- Solution 1: Since "Other Brands" is 20% of the pie, then 900 is 20% of the pie. So, 100% of the pie is 900 ÷ 20% = 4,500. Select (E) 4,500.
- Solution 2:
- Estimate the answer: The slice of the pie marked "Other Brands" consists of 900. Looking at that slice of the pie as compared with the entire pie, the entire pie should consist of somewhat more than 4,000.
- Look at the answer choices: The only answer greater than 4,000 is (E) 4,500. Select that answer.

- A rectangular floor that is 12 feet by 18 feet is 4 yards by 6 yards. So, 4 × 6 = 24 yards of carpeting will be required. Select (C) 24.
- Estimate the answer: Since the puppy must weigh more than either the kitten or the bunny, the puppy must weigh more than ½ of 9, so probably around 5 or 6 pounds.
- Look at the answer choices: (D) 5 pounds and (E) 6 pounds seem reasonable. Eliminate the other three possibilities.
- Guess and check: Try (D) and (E) and see which one works with all three weighings. If the puppy weighs 5 pounds, then the bunny weighs four pounds and the kitten weighs 3 pounds, so the kitten and the bunny together weigh 7 pounds, which is correct. Select (D) 5 pounds.

- Estimate the answer: 40 feet is somewhat more than double 16 feet, so the answer should be somewhat more than 2(¼) = ½ inch.
- Look at the answer choices: Answers (B) and (C) are both somewhat more than ½. The other three answers make no sense, so eliminate them.
- This proportion can be solved using either arithmetic or algebra. Using arithmetic, lthe length = (40 feet)(¼ inch/16 feet) = 5/8 inch. Select (B) 5/8.

- Substitute (
`p`, 0) into both equations:0 = −`p`² + 9 (Equation 1)

0 =`p`² - 9 (Equation 2) - Subtract Equation 2 from equation 1:
0 = −2Select (A) 3.
`p`² + 18`p`² = 9`p`= 3

- Substitute (
- Estimate the answer: The two machines make 750 bolts per hour. To make 900 bolts, it will take slightly more than 1 hour, but not too much more (less than 1.5 hours).
- Look at the answer choices: The only answer slightly more than 1 hour is (B) 72 [minutes]. Select that answer.

- Solution 1: The function
`g`(`t`) decreases by 2 whenever`t`increases by 1, so its slope must be −2. The function is equal to 2 when`t`is equal to 0, so its`y`-intercept must be 2. An equation of a line with slope −2 and`y`-intercept 2 is`g`(`t`) = −2`t`+ 2. Select (E)`g`(`t`) = −2`t`+ 2. - Solution 2:
- Look at the answer choices: When
`t`= 0,`g`(`t`) = 2. Eliminate all answers for which`g`(`t`) is not 2 when`t`= 0. We can eliminate answers (A), (B), and (C). - Look at the answer choices: When
`t`= 1,`g`(`t`) = 0. Looking at (D) and (E), eliminate whichever does not equal 0 when`t`= 1. (D) equals 1, so eliminate it and select (E)`g`(`t`) = −2`t`+ 2.

- Look at the answer choices: When

- Solution 1: The function
- Look at the answer choices: :
- Looking at the graph, there are two students who travel 2 miles, so (A) is false.
- There seems to be a fair number of students who travel 4 or more miles to school than there are who travel 3 or less, so (B) is false.
- three 12th graders travel 6 or more miles to school, while only two 11th graders do, so (C) is true. Select (C) More 12th graders than 11th graders travel 6 or more miles to school.

- List the numbers out. There are 10 such numbers: 304, 314, 324, 334, 344, 354, 364, 374, 384, and 394. Select (A) 10.
- Estimate the answer: If
`y`=`mx`+`b`has a slight downward slope,`y`= −3`mx`+ b will have a steep upward slope. Because the`y`-intercept of the two graphs is the same, it will intersect the`y`-axis at`x`= −1. - Look at the answer choices: The only graph with a steep rightward slope that intersects the
`y`-axis at −1 is (D). Select that answer.

- Estimate the answer: If
- Draw a diagram: It may be helpful to draw a diagram of the cube.
- If the volume of a cube is 8, then the length, width, and depth of the cube must each be 2. So, the shortest distance from the centre of the cube to the base is half of the side length, or 1. Select (A) 1.

- Try a special case: Say that
`x`= 1 and`z`= 1. Then`y`= 5. If we double both`x`and`z`, then`y`= 5(2³)/2 = 20. So,`y`is multiplied by 4. - Look at the answer choices: The only answer that makes sense is (E)
`y`is multiplied by 4. Select that answer.

- Try a special case: Say that
- Guess and check: Try the given answers until you find one that works:
- For (C) Three,
`V`(3) = 5000(4/5)³ = $2,560. That's too low, so try a smaller number. - For (B) Two,
`V`(2) = 5000(4/5)² = $3,200. Select (B) Two.

- For (C) Three,
- Read the question and understand what it is asking: Another way of looking at the problem is the following: Starting with the symbols ABC, alternately switch the first two symbols and then the last two symbols, resulting in
`ABC`,`BAC`,`BCA`, ... - Continuing that pattern, we get:
`ABC`(Start),`BAC`(Step 1),`BCA`(Step 2),`CBA`(Step 3),`CAB`(Step 4),`ACB`(Step 5),`ABC`(Step 6). The original order is first repeated at step 6. Select (D) 6.

- Read the question and understand what it is asking: Another way of looking at the problem is the following: Starting with the symbols ABC, alternately switch the first two symbols and then the last two symbols, resulting in
- Try a special case: Say that the numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. Then, the median is 6.
- Look at the answer choices: Look at the answers and see what happens with our special case:
- If each number is doubled, then the median would be 12. (A) is wrong.
- If each number is increased by 10, then the median would be 16. (B) is wrong.
- If the smallest number (1) were increased by 6 or more, then the new median would be 7. (C) is wrong.
- If the largest number (11) were decreased by 6 or more, then the new median would be 5. (D) is wrong.
- If the largest number were increased, the median would still be 6. This must be the correct answer. Select (E) Increasing the largest number only.

- This is a good one. Two solutions are presented here, one more formal and one informal. If you can find either, you are well on your way to an 800:
- Solution 1:
- Draw a diagram: Draw line
`RB`on the diagram given. - Since both
`RB`and`AC`are diagonals of rectangle`ABCR`, they must be equal. Because`RB`is a radius, it must have length 6, so`AC`also has length 6. - Now, the distance from
`S`to`A`to`R`to`C`to`T`is 12, because`SR`and`RT`are radii of the circle and are both equal to 6. Now, the distance from`A`to`R`to`C`is 8, because we are told that the length plus width of the rectangle is 8. Subtracting the two lengths, the distance from`S`to`A`plus the distance from`C`to`T`is 12 − 8 = 4. - Finally, because angle
`SRT`is a right angle, arc`ST`is ¼ of the circumference of a circle with radius 6, or ¼(2π(6)) = 3π. So, the total perimeter of the shaded region is 6 + 4 + 3π = 10 + 3π. Select (B) 10 + 3π.

- Draw a diagram: Draw line
- Solution 2: To find the perimeter, there are four lengths we need to find: arc
`TS`,`SA`,`AC`, and`CT`.- Because angle
`SRT`is a right angle, arc`ST`is ¼ of the circumference of a circle with radius 6, or ¼(2π(6)) = 3π. - Because the diagram is drawn to scale, we can estimate the length of
`CT`. It appears to be about half of`RT`. As`RT`is 6,`CT`is around 3. - We are told that the length plus width of rectangle
`ABCR`is 8. So, if we've estimated`CT`to be about 3, then`RC`must be about 3,`AR`must be around 5, and`SA`must be around 1. `AC`is the hypotenuse of a right triangle with legs`AR`and`RC`. Since`AR`must be around 5, and`RC`is around 3, by the Pythagorean theorem (see Reference Information)`AC`is somewhere around √3² + 5², or somewhere around 6.- In total, the length is probably somewhere around 10 + 3π
- Look at the answer choices: Based on our estimate, the only reasonable answer is (B) 10 + 3π. Select that answer.

- Because angle
- Solution 3:
- Estimate the answer: If you aren't able to come up with either of the two solutions, you might be able to eyeball the diagram and determine that
`AC`looks like it's around 6,`CT`looks like it's around 3, and`SA`looks like it's around 1. Since the length of arc`ST`is 3π (see above for calculations), the perimeter is around 10 + 3π. - Look at the answer choices: The only answer that makes sense is (B) 10 + 3π. Select that answer.

- Estimate the answer: If you aren't able to come up with either of the two solutions, you might be able to eyeball the diagram and determine that

- Solution 1: