Math Lair Home > Test Preparation > **Solutions for Practice Test 8, The Official SAT Study Guide, Section 3**

*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 3 of practice test #8 in The Official SAT Study Guide, second edition, found on pages 830–835. 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.

- Look at the answer choices: 1/5 is 0.2, while ¼ is 0.25. The only answer in between these two values is (D) 0.21. Select that answer.
- Draw a diagram: You may find it helpful to draw a diagram and plot the five points given on it.
- Look at the answer choices: Answer (B) is 0 units away from the origin in the
`x`-direction and ½ units away in the`y`-direction. Both coordinates, in absolute terms, are the smallest (or tied for smallest) of all five answers. Select (B) (0, ½).

- Estimate the answer: The angle labelled
`y`° is greater than a right angle, but not significantly so; perhaps it is around 110° or so. - Look at the answer choices: There are a few answers around 110°. The angle is almost certainly less than 135°, so we can eliminate answer (E), but the other four possibilities seem possible.
- The angle labeled
`y`° is opposite to an angle whose measurement is 3`x`°. Since opposite, or vertical, angles are equal,`y`= 3`x`. - Because a straight angle in the diagram is comprised of five angles each of whose measurement is
`x`°, then`x`= 180/5 = 36. So,`y`= 3 × 36 = 108. Select (A) 108.

- Estimate the answer: The angle labelled
- Estimate the answer: Since 65
`x`must be around 6,500,`x`must be around 100. - Look at the answer choices: We can eliminate answers (A), (B), and (E). We still need to decide between (C) 100 and (D) 101.
- Solve the equation:6,565 = 65(Select (C) 100.
`x`+ 1)

6,565 = 65`x`+ 65

6,500 = 65`x``x`= 100

- Estimate the answer: Since 65
- If
`m`^{x}·`m`^{7}=`m`^{28}, then`x`= 21. If (`m`^{5})^{y}=`m`^{15}, then`y`= 3. 21 + 3 = 24, so select (D) 24. - Estimate the answer: Between 1987 and 1990, the graph appears to show a decline of slightly over 40,000 in three years, so the answer appears to be slightly over 40,000/3 per year, say around 13,500.
- Look at the answer choices: The only answer slightly over 13,500 is (C) 14,000. Select that answer.

- Estimate the answer: The diagram is not drawn to scale, but it doesn't appear to be too inaccurate, so we may be able to get a rough guess from it. The length of
`CE`appears to be somewhat longer than the length of`BD`; if the length of BD is 8, perhaps the length of`CE`is around 10. - Look at the answer choices: Answers (B) and (C) seem reasonable. The other answers could probably be eliminated.
- Work backwards: We could find the length of
`CE`if we knew the lengths of`CB`and`BE`. But how can we find either length? Well, if`x`=`y`, then the lower triangle is a 45°-45°-90° triangle, and, because angle`CBD`must also equal`y`°, the upper triangle is also a 45°-45°-90° triangle. So (and look at the Reference Information if necessary) if the length of`AB`is 4, then the length of`BE`is 4/√2 = 2√2, and the length of`CB`is 8/√2 = 4√. So, the length of `CE`is 4√2 + 2√2 = 6√2. Select (B) 6√2 (approximately 8.49).

- Estimate the answer: The diagram is not drawn to scale, but it doesn't appear to be too inaccurate, so we may be able to get a rough guess from it. The length of
- Try a special case: Say that the price of coffee beans is 16 dollars for 8 ounces, and each ounce makes 1 cup of coffee. Then, the cost of the coffee beans required to make 1 cup of coffee is $2. In other words, if
`d`= 16 and`c`= 1, then the answer is 2. - Look at the answer choices: See which answers result in $2 when
`d`= 16 and`c`= 1:- (A) evaluates to 2
- (B) evaluates to 2
- (C) evalutes to 0.5
- (D) evalutes to 128
- (E) evaluates to 128

- Try a special case: Say that the price of coffee beans is 16 dollars for 8 ounces (
`d`= 16), and that each ounce makes 2 cups of coffee (`c`= 2). Then, the cost of coffee beans per cup of coffee is $1. - Look at the answer choices: See which of (A) and (B) evaluates to $1 when
`d`= 16 and`c`= 2:- (A) evaluates to 1
- (B) evaluates to 4

`d`/8`c`.

- Try a special case: Say that the price of coffee beans is 16 dollars for 8 ounces, and each ounce makes 1 cup of coffee. Then, the cost of the coffee beans required to make 1 cup of coffee is $2. In other words, if
- Cross-multiply, giving
`ab`= 120. Enter**120**. - If each term after the first is 1/5 of the term preceding it, the 4th term will be 6/5 and the 5th term will be 6/25. Enter
**6/25**. - Draw a diagram: Draw each piece of information in turn, starting with
`AB`, then`C`, and then`D`, which should give you something like the following: - The distance between
`A`and`D`is 6. There are a few places where`E`could be located; one place is one unit to the right of`A`and five units to the left of`D`. Enter**1**.

- Draw a diagram: Draw each piece of information in turn, starting with
- Solution 1: If the sum of five consecutive integers is 185, then the middle integer must be 185/5 = 37. Then, the largest integer is 39. Enter
**39**. - Solution 2: Let
`x`be the largest of the 5 consecutive integers:Enter`x`+ (`x`− 1) + (`x`− 2) + (`x`− 3) + (`x`− 4) = 185

5`x`− 10 = 185

5`x`= 195

`x`= 39**39**. - Solution 3: Guess and check: Say that the smallest number is 30. Then 30 + 31 + 32 + 33 + 34 = 160, which is 25 too small. If we add 5 to each number, we get 35 + 36 + 37 + 38 + 39 = 185, which is the correct answer. The greatest number in this sum is 39, so enter
**39**.

- Solution 1: If the sum of five consecutive integers is 185, then the middle integer must be 185/5 = 37. Then, the largest integer is 39. Enter
- Let
`x`represent the amount of sales for the month:1200 + 0.2Enter`x`= 2500

0.2`x`= 1300`x`= 6500**6500**. - From the Reference Information, the number of degrees of arc in a circle is 360. So, a wedge consisting of 40 degrees of arc is 40/360 = 1/9 of a circle. If the entire circle weighs 2.5 grams, the wedge weighs (1/9)(2.5) = 5/18 gram. Enter
**5/18**. - Factor the first equation and substitute the second equation in:Enter
`x`² −`y`² = 10

(`x`+`y`)(`x`−`y`) = 10

5(`x`−`y`) = 10`x`−`y`= 2**2**. - Work backwards: We could find the area of the shaded area if we knew the area of the larger square and the area of the four triangles. Now, the area of the larger square is 3 × 3 = 9, and the area of each triangle is ½(2)(1) = 1. So, the area of the shaded area is 9 − 1 − 1 − 1 − 1 =
**5**. - Read the question and understand what it is asking: You are asked to find a number
`k`such that 13 ÷`k`leaves remainder 2. - Guess and check: Try values for
`k`until you find a match. 11 seems like a good place to start, because it is 2 away from 13. In fact, 13 ÷ 11 = 1 remainder 2, so 11 solves the problem. Enter**11**.

- Read the question and understand what it is asking: You are asked to find a number
- If the average of the class of
`p`students is 70, then the total number of marks in the first class is 70`p`. If the average of the class of`n`students is 92, then the total number of marks in that class is 92`n`. The total number of marks in the two classes is 86(`p`+`n`). We can use this information to set up an equation:70Enter`p`+ 92`n`= 86(`p`+`n`)

6`n`= 16`p``p`/`n`= 6/16**6/16**.