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

Here are solutions for section 4 of the fifth practice test in The Official SAT Study Guide, second edition, found on pages 650–655. 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. If s + t = 3, then s + t − 6 = 3 − 6 = −3. Select (A) −3.
2. Look at the diagram for a point that appears to be either much closer to P or much closer to Q. C is such a point. Select (C) C.
3. • Estimate the answer: Web sites represent slightly over 75%, so more than ¾ of the circle should represent web sites. E-mail represents significantly more than either news groups or other, and news groups and other are equal. So, our desired answer should show web site taking over ¾ of the circle, e-mail taking most of the rest of the circle, and news groups and other taking small but equal fractions of the circle.
   • Look at the answer choices: Only (D) matches our estimate. Select that.
4. Solution 1: Guess and check: Look for a fraction equal to ¾ with a numerator 5 less than the denominator. ¾ = 6/8 = 9/12 = 12/16 = 15/20. 15/20 is equal to ¼ and has a numerator 5 less than the denominator. Its denominator is 20, so select (D) 20.
   Solution 2:
   • Let x represent the numerator of the desired fraction, and set up an equation:
     x / (x + 5) = ¾
   • Cross-multiply and solve the equation:
     4x = 3x + 15
     x = 15
   • If the numerator of the fraction is 15, then the denominator must be 20. Select (D) 20.
5. Solution 1:
   • Estimate the answer: The height of the triangle is 4, the base is 5k − 2k = 3k, and the area is 18, then k must be somewhere around 2 or 3 or 4.
   • Look at the answer choices: The only answer choice within range of our estimate is (E) 3. Select that answer.
   Solution 2:
   • Estimate the answer as above.
   • The area of a triangle is ½bh, where b is the base and h the height. The height of the triangle is 6 − 2 = 4, and the base is 5k − 2k = 3k, and the area is 18. So:
     18 = ½(4)(3k)
     18 = 6k
     3 = k
     Select (E) 3.
6. Solution 1:
   • Read the question carefully and determine what it is asking: You are asked to find the value of 1/m in terms of k.
   • Try a special case: Say that k = 2. The equation becomes:
     5m² = 100m
   • Solve the equation for 1/m. To do that, we'll divide both sides by 100m²:
     (1/20) = 1/m
   • Look at the answer choices: Determine which answer choice is equal to 1/20 when k = 2. The only such choice is (D) 1/10k. Select that answer. Note: If you had tried a special case of, say, k = 1, several choices would have matched. In that case, just eliminate the choices that didn't match and try another special case.
   Solution 2:
   • Read the question carefully and determine what it is asking: You are asked to find the value of 1/m in terms of k.
   • Solve the given equation for 1/m:
     10m²k⁻¹ = 100m
     mk⁻¹ = 10
     m = 10k
     1/m = 1/10k
     Select (D) 1/10k.
7. • Read the question and understand what it is asking: Edna walks east a distance of 4 × 4 = 16 kilometres. Nancy walks north a distance of 3 × 4 = 12 kilometres. Determine how far away they are from one another.
   • Draw a diagram: Draw a diagram of the situation. It should resemble the following:
     
   • The triangle is a right triangle, so we can use the Pythagorean theorem (consult the Reference Information if necessary). The Pythagorean theorem states that, for a right triangle, c² = a² + b², where c is the hypotenuse and a and b the two legs. For this triangle, we have:
     c² = 12² + 16²
     c² = 144 + 256
     c² = 400
     c = 20
     Select (E) 20.
8. • Draw a diagram: The diagram is already drawn, but circle the point on the graph representing x = 3.
   • If f(b) = f(3), then b will be directly left or right of the point we just circled. Looking at the graph, this point is at x = −1. Select (C) −1.
9. The family needs 5 × 4 = 20 bottles of water. If the water is sold only in 3-bottle packages, they will need 20/3 = 6.666... packages. However, they can't buy .666... of a package, so round up to the nearest integer, 7. Enter 7.
10. Guess and check:
    • One value that obviously solves the first equation is k = 7. However, it does not solve the second, because |7 − 5| ≠ 8.
    • One value that obviously solves the second equation is k = 13. It also solves the first equation, because |10 − 13| = 3. Enter 13.
11. • Estimate the answer: The value of x is definitely greater than 90. It appears to be about 120 or so.
    • Draw a diagram: A diagram is given. The value of x is not immediately obvious, so start by filling in what you can determine. Because line m makes a right angle with the vertical line, and because the angle between the vertical line and l is 65°, then the angle between l and m is 90 − 65 = 25°. Fill that in on both the upper right and lower left sides of the diagram. Now, looking at the diagram, the lower angles 25°, 20°, and x° must add to 180°. So:
      25 + 20 + x = 180
      x = 135
      Enter 135.
12. • Draw a diagram: Draw a set of nine blanks:
      __ __ __ __ __ __ __ __ __
    • You are told that the median is 42. The median is the middle value, so enter that in the middle blank:
      __ __ __ __ 42 __ __ __ __
    • You are told that the integers are consecutive, so enter consecutive integers in the blanks. We are only asked for the largest integer, so just enter numbers starting after the median:
      __ __ __ __ 42 43 44 45 46
      The largest of these integers is 46.
13. If 2f(p) = 20, then:
    2f(p) = 20
    2(p + 1) = 20
    p + 1 = 10
    p = 9
    So, f(3p) = (3p + 1) = (3(9) + 1) = 28.
14. • Because ∠LMN is supplemental to an angle labeled 125°, the value of ∠LMN is 180 − 125 = 55°.
    • Because LMN is a triangle and the lengths of LM and LN are equal, the angles opposite those sides are equal as well, so ∠LNM = ∠LMN = 55°.
    • Triangles JKN and JLM are similar, because they have two angles the same. So, ∠KNJ = ∠LMJ = 55°.
    • Finally, x = 180 − 55 − 55 = 70. Enter 70.
15. The amount of orange juice added in the mixture is (4/5)(1/3) = 4/15 cup. There was 1/5 cup of orange juice to begin with, so the total amount is 4/15 + 1/5 = 7/15. Enter 7/15.
16. Convert the sentence into an equation:

    If	[ignore]
    a	a
    +	+
    2b	2b
    is equal to	=
    125 percent	125%
    of	·
    4b	4b

    Putting it all together, we get:
    a + 2b = 125% · 4b
    Solving this equation for a/b, we get:
    a + 2b = 125% · 4b a + 2b = 5b a = 3b a/b = 3
    Enter 3.
17. If there are 9 equal intervals between 0 and 1 on the number line, the one labelled √x is equal to 6/9, or ⅔. So, if √x = ⅔, then x = 4/9. Enter 4/9.
18. • Draw a diagram: Draw a diagram of the graph:
      
    • Use the Pythagorean Theorem (see Reference Information if required) to find the length of the third side of the triangle. It is equal to √17² − 15² = √64 = 8. So, the value of x must be 8 away from 10. One possibility is 2. Enter 2.