Solutions for Practice Test 4, The Official SAT Study Guide, Section 6

Here are solutions for section 6 of the fourth practice test in The Official SAT Study Guide, second edition, found on pages 581–586. The following solutions 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.

1. Solution 1:
   • First, convert the right-hand side to an improper fraction, namely 15/2.
   • So, if (3 + ♢)/2 = 15/2, then
     3 + ♢ = 15
     ♢ = 12
     Select (D) 12.
   Solution 2: Guess and check. Try each answer, starting with the middle answer, to see if it works. Does (C) 9 work? (3 + 9)/2 = 6, which is not 7½. So, try a larger answer, say (D). (3 + 12)/2 = 15/2 = 7½, so this is the correct answer. Select (D) 12.
2. Solution 1: From the diagram, angle 2 is an obtuse angle and angle 4 is another obtuse angle. So, look for two obtuse angles formed by m. Angles 6 and 8 are obtuse. Therefore, the answer is (D) 6 and 8.

   Solution 2: Using Z, F, and C patterns (or however you remember congruent angles when a line intersects a pair of parallel lines), you can determine that angle 2 is congruent to angles 6 and 8, and angle 4 is congruent to angles 6 and 8. Based on this information, the only answer that makes sense is (D) 6 and 8. Select that answer.

3. There are a few ways of solving this problem; I'll only illustrate the most direct route.
   • Work backwards: We could determine the total number of unemployed women if we knew the number of employed women (since we know the total). We have the number of employed people and the number of employed men, so we can find the number of employed women.
   • The number of employed women is equal to (total employed - total men employed), or 48,000 − 27,000 = 21,000. The number of unemployed women is equal to (total number of women - total number of women employed) or 21,500 − 21,000 = 500. Select (A) 500.
4. Substitute k = 4 in A(k) = 4k − 30:
   A(15) = 4(15) − 30
   A(15) = 30
   Select (E) $30.
5. Solution 1: Substitute xr = v in v = kr:
   xr = kr
   x = k
   So, x is equal to k. Select (D) x.

   Solution 2:

   • Try a special case: Pick specific numbers for x and r (say x = 2 and r = 3). Since xr = v, then v = 2(3) = 6. Now:
     v = kr
     6 = 3k
     k = 2
   • Look at the answer choices: Substitute x = 2 in each of the answer choices and see which evaluates to 2, the value of k we found in the previous step. It's pretty obvious that the only such choice is (D) x. Select that answer.
6. • Read the question carefully and understand what it is asking: You have a basket of eggs. For every two eggs in the basket, there are three brown eggs. It is not possible to have every number of eggs in the basket; you are asked to find which of the choices represents a number you can't have.
   • Draw a diagram: You may find it helpful to draw a diagram of a basket of eggs, with white and brown eggs in the ratio of 2 to 3.
   • Look at the answer choices: If, for every two white eggs in the basket, there are three brown eggs, there number of eggs must be a multiple of (2 + 3) = 5. The only answer choice not a multiple of 5 is (B) 12. Select that answer.
7. • Estimate the answer: It's difficult to come up with an exact estimate; however, To find r, we would multiply 18 by some number (say x) and, to find t, we could divide 18 by x². So, r has to be greater than 18, and t has to be a fair bit less than 18. So, the answer has to be more than 18 and less than 18 × 18 = 324.
   • Look at the answer choices: The answer can't be (A) 18 or (E) 324. Eliminate those three choices. There are still three choices to choose from.
   • 18√18 can be written as 18√9√2. Since √9 = 3, the expression becomes 54√2, which is what we're looking for. To find rt, multiply 54 and 2: 54 × 2 = 108. Select (C) 108.
8. Solution 1: This question is pretty easy if you notice that the three triangles on the left-hand side form a quadrilateral, and you remember that the interior angles of a quadrilateral sum to 360°. The four angles of the quadrilateral are:
   • a°
   • b° + b°
   • a° + c°
   • b°
   The sum of these must be equal to 360°:
   a° + b° + b° + a° + c° + b° = 360°
   2a + 3b + c = 360
   c = 360 − 2a −3b
   Select (E) 360 − 2a −3b.

   Solution 2: If the above solution didn't occur to you, it's still possible to solve the problem; it just takes a bit longer.

   • You are told in the Reference Information that "The sum of the measures in degrees of the angles of a triangle is 180." There are two triangles with angles of a° and b°. The third angle of each must be (180 − a − b)°.
   • This information allows us to find the third angle in the triangle with c, as that angle plus the two angles we found form a straight angle (180°). Calling this angle x°, we have:
     (180 − a − b)° + (180 − a − b)° + x° = 180°
     x° = (2a + 2b − 180)°
   • Finally, find the value of c, being very careful with the negative signs:
     x° + b° + c° = 180°
     (2a + 2b − 180)° + b° + c° = 180°
     c = 360 − 2a −3b
     Select (E) 360 − 2a −3b.
9. Don't try to solve for t. Instead, just multiply both sides of the equation by 4:
   4t³ = 1404
   Enter 1404.
10. • Draw a diagram: You may find it helpful to draw a number line to visualize the problem.
    • Looking at the number line or thinking about the problem, the difference between 53 and 62 is 9. So, the number we're looking for will be a distance of 4.5 from both. The answer is 53 + 4.5 = 62 − 4.5 = 57.5. Enter 57.5.
11. • Draw a diagram: It may be helpful to draw a diagram of a triangle with two equal angles and/or sides of 50 and 30.
    • If a triangle has two angles that are equal and one angle not equal to the other two, it must have two sides that are equal and one side not equal to the other two. So, the third side must be equal to one of the other two sides—it must be either 30 or 50. The least possible value for the perimeter would occur when the third side is 30. So, the perimeter is 30 + 30 + 50 = 110. Enter 110.
12. It's possible to solve this problem in a more intuitive manner, but I've chosen to include a solution with all of the steps.
    • Estimate the answer: Based on the numbers in the question, the answer looks like it should be an integer. If x² + y² = 77, then x² > 77, then x² > 77, so x is probably greater than or equal to 9. If x + y = 11, then x is probably no more than 11. This is helpful information; our estimate suggests x must be one of 9, 10, or 11, so if we can't find the answer we can make an educated guess of one of these three numbers.
    • Looking at the first equation:
      x² −y² = 77
      (x + y)(x − y) = 77
      Substituting the second equation (x + y = 11) into the above, we get:
      11(x − y) = 77
      x − y = 7
      We now have two linear equations in two variables: x − y = 7 and x + y = 11. To solve for x, add them together:
      2x = 18
      x = 9
      Enter 9.
13. Solution 1: Try a special case: Say that Tameka cut the pieces so that the angle is 24°. There would be 360/24 = 15 pieces. So, 15 is a possible answer. Enter 15.

    Solution 2: Consider a different problem: Say that we were asked to find the number of pieces if Tameka cut the pizza into slices with a 30° angle. There would then be 360/30 = 12 pieces. However, the actual question specifies that the angle is less than 30°, so there must be more than 12 pieces. Enter the next integer higher than 12, which is 13.

14. Solution 1: Guess and check: Say that a = 1. The sum of the terms would be 1 + 3 + 9 + 27 + 81 = 121. This isn't right. More importantly, though, it is 121/605 = 1/5 of the correct answer.
    Since it appears that the value should be 5 times larger, try a = 5. 5 + 15 + 45 + 135 + 405 = 605. This answer is correct. Enter 5.
15. • Draw a diagram: A diagram is given, but it is not drawn to scale. The diagram doesn't look too bad, but if you don't feel it's accurate, you may find it helpful to draw, on the diagram, points S and T based on what the question says.
    • Estimate the answer: Looking at the diagram, triangle PST seems to be about ½ of the area of PQR. So, we should expect an answer around ½, and, if we can't answer the question, we should guess something around there.
    • The area of triangle PST is ½bh = ½(PT)(SV).
    • The area of triangle PQR is ½bh = ½(PR)(QV).
    • The ratio of the area of PST to PQR is:
        ½bh = ½(PT)(SV) ÷ ½(PR)(QV)
      = (PT)(SV) ÷ ½(PR)(QV)
      = (PT/PR)(SV/QV)
      = ¾(SV/QV)
      We are not given SV/QV, but SV = QV − QS. So:
      = ¾((QV − QS)/QV)
      = ¾(QV/QV − QS/QV)
      We are given QS/QV (= ⅓), so:
      = ¾(1 − ⅓)
      = ¾(⅔)
      = ½
      Enter 1/2.
16. • First, find h(2m):
      h(2m) = 14 + (2m)²/4
      h(2m) = 14 + m²
      The question states that the result above is equal to 9m, so:
      14 + m² = 9m
      m² − 9m + 14 = 0
      (m − 2)(m − 7) = 0
      m = 2 or m = 7
      Enter 2 (or, 7).
17. • Read the question and understand what it is asking: Several clocks are going to chime various times at 7:30, at 8:00, and at 8:30. We need to find the total number of chimes.
    • Estimate the answer: There are 15 clocks that are going to chime 8 times at 8:00 am, and there are several other clocks chiming at other times. Perhaps a rough estimate of the number of chimes might be 15 × 10 = 150.
    • Go through each checked cell in the table and determine the number of chimes it represents. Keep in mind there are two half-hours as well as 8:00 in the time period in question. The results are shown in the table below:

      10 × 8 = 80		10 × 2 = 20
      5 × 8 = 40
      	3 × 1 = 3	3 × 2 = 6

      Summing the numbers in the table together:
      80 + 20 + 40 + 3 + 6 = 149
      Enter 149.
18. Solution 1: There are three different places in which the shaded card can go. Having placed the shaded card, there are four different places where the next card can go, three different places for the card after that, two different places for the card after that, and one place for the final card. This makes 3 × 4 × 3 × 2 × 1 = 72 possible arrangements. Enter 72.

    Solution 2: To find the number of arrangements, find the total number of arrangements if there were no conditions, and subtract the number of arrangements where the shaded card is at either end. If there were no conditions, there would be 5! = 120 arrangements. However, there are 4! = 24 ways to arrange the cards so that the shaded card is on the left end, and 24 ways to arrange the cards so that the shaded card is on the right end. So, the total number of valid possibilities is 120 − 24 − 24 = 72. Enter 72.