SAT MATH Dr. John Chung 58 Perfect Tips Designed to help Students get a Perfect Score on the SAT
Tip 1 β Absolute Value β’ The absolute value of x, π₯ , is regarded as the distance of x from zero. β’ How do we convert the general interval into an expression using absolute value? β’ Example: 10 β€ π₯ β€ 30 β Step 1) Find the midpoint:
10+30 2
= 20
β Step 2) Find the distance to either point: 20 β 10 = 10 β Step 3) Substitute: π₯ β ππππππππ‘ β€ πππ π‘ππππ
Tip 1 β Absolute Value (cont.) β’ If π₯ + 3 < 5, what is the value of x?
β’ If π₯ + 3 > 5, what is the value of x?
Tip 1 β Absolute Value (cont.) β’ At a bottling company, a computerized machine accepts a bottle only if the number of fluid ounces is greater than or 3 4 equal to 5 and less than or equal to 6 . If the machine 7 7 accepts a bottle containing π fluid ounces, which of the following describes all possible values of π? A.
πβ6 <
B.
πβ6 <
C.
π+6 >
D.
6βπ β€
E.
π+6 β€
4 7 3 7 4 7 4 7 4 7
Tip 1 β Absolute Value (cont.) β’ At a milk company, Machine X fills a box with milk, and machine Y eliminates the milk-box if the weight is less than 450 grams, or greater than 500 grams. If the weight of the box that will be eliminated by machine Y is E, in grams, which of the following describes all possible values of E? A. B. C. D. E.
πΈ β 475 πΈ + 475 πΈ β 500 475 β πΈ πΈ β 475
< 25 < 25 > 450 = 25 > 25
Tip #2 - Ratio to Similar Figures β’ Two polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are in proportion β’ If the ratio of the corresponding lengths is a:b, then the ratio of the areas is π2 : π 2 and the ratio of the volumes is π3 : π 3 .
Ratio to Similar Figures (cont) β’ The ratio of the sides of 2 similar triangles is 5:2. If the area of the larger triangle is 30, what is the area of the smaller triangle? β’ Solution β The ratio of areas is 25:4 β 25k = 30 or k = 1.2 β Therefore 4k = 4(1.2) = 4.8
Ratio to Similar Figures (cont.) β’ In Triangle ABC, AB, PQ, & RS are parallel and the ratio of the lengths is AQ:QS:SC = 2:2:3. If the area of quadrilateral PRSQ is 48, what is the area of Triangle ABC? A. B. C. D. E.
84 92 105 144 147
Tip 3: Combined Range of Two Intervals Rules
Example 1
If 5 β€ π΄ β€ 10 πππ 2 β€ π΅ β€ 5,
β’ Given 2 β€ π β€ 8 πππ 1 β€ π β€ 4. By how much is the π maximum of > the
i. ii. iii. iv.
7 β€ π΄ + π΅ β€ 15 10 β€ π΄ Γ π΅ β€50 0β€π΄βπ΅ β€8 1β€
π΄ π΅
β€5
Smallest value β€ Combined Range β€ Largest Value
minimum of
π π ? π
8 1
i.
Solution: max is = 8
ii.
Min is = = .5
2 4
1 2
iii. 8 β .5 = 7.5
Tip 3: Combined Range of Two Intervals (cont.) β’ If β2 < π₯ < 4 πππ β 3 < π¦ < 2, what are all possible values of π₯ β π¦? A. B. C. D. E.
β4 < π₯ β π¦ < 2 1<π₯βπ¦ <7 1<π₯βπ¦ <4 β4 < π₯ β π¦ < 7 β5 < π₯ β π¦ < 7
Tip 3: Combined Range of Two Intervals (cont.) β’ The value of p is between 1 and 4, and the value of q is between 2 and 6. Which of the following is π a possible value of ? π
A. Between B. Between
C. Between D. Between E. Between
1 2 πππ 2 3 2 πππ 2 3 1 πππ 6 2 1 2 πππ 2 1 1 πππ 1 2 2
Tip 4: Classifying a Group in Two Different Ways β’ Organize the information in a table and use a convenient number β Example: In a certain reading group organized of only senior and junior students, 3/5 of the students are boys, and the ratio of seniors to juniors is 4:5. If 2/3 of girls are seniors, what fraction of the boys are juniors? Seniors Juniors
BOYS
GIRLS
4 4 8 β = 9 15 45 3 8 19 β = 5 45 45 3/5
2 2 4 β = 3 5 15
4/9 5/9
2/5
1
Tip 4: Classifying a Group in Two Different Ways (cont.) β’ On a certain college faculty, 4/7 of the professors are male, and the ratio of the professors older than 50 years to the professors less than or equal to 50 years is 2:5. If 1/5 of the male professors are older than 50 years, what fraction of female professors are less than or equal to 50 years. > 50
Males
Females
1 4 β 4 = = 0.8 5 5
4 2 β = 1.2 5 3 β 1.2 = 1.8
2
3
7
β€ 50 4
5
Tip 5: Direct Variation β’ When 2 variables are related in such a way that π¦ = ππ₯, the 2 variables are said to be in direct variation. β’ Expression of direct variation: i. π¦ = ππ₯ π¦ ii. π₯ = π
β’ Geometric interpretation: π¦ = ππ₯ is a special linear equation where the π¦ β πππ‘ππππππ‘ is (0, 0). β’ In the π₯π¦ β coordinate plane, π¦ = ππ₯, where π ππ π ππππ, but π¦ β πππ‘ππππππ‘ must be zero.
Tip 5: Direct Variation (cont.) β’ The value y changes directly proportional to the value of x. If y = 15 when x = 5, what is the value of y when x = 12.5?
Tip 5: Direct Variation (cont.) β’ The 2 adjoining triangles are similar. What is the length of side DF?
12
β 6 β
82 13 90 13
β β 8
β
100 13
5
18
Tip 6: Inverse Variation β’ When 2 variables are related in such a way that π₯π¦ = π, the two variables are said to be in inverse variation. β Properties: β’ The value of two variables change in an opposite way; that is, as one variable increases, the other decreases. β’ The product π is unchanged
Tip 7: Special Triangles β’ Angle-based Right Triangle β 30-60-90 triangle β’ In a triangle whose angles are in the ratio 1:2:3, the sides are in the ratio 1, 3, 2
β 45-45-90 Triangle β’ In a triangle whose 3 angles are in the ratio 1:1:2, the sides are in the ratio 1, 1, 2
β’ Side-based triangles β’ Right triangles whose sides are Pythagorean triples as follows. 3:4:5
5:12:13
8:15:17
7:24:25
9:40:41
11:60:61
Tip 7: Special Triangles (cont.) β’ An equilateral triangle ABC is inscribed inside a circle with radius = 10. β What is the area of βABC?
B
A
C
Tip 7: Special Triangles (cont.) β’ Figure ABDE is a square and βBCD is an equilateral triangle. If the area of βBCD is 16 3 β What is the area of the square? A. 32 B. 32 3 C. 64 D. 64 2 E. 72
Tip 8: Exponents β’ The exponent is the number of times the base is used as a factor. 5 = πππ π 52 = 25 2 = ππ₯ππππππ‘ 25 = πππ€ππ
β’ The mathematical operations of exponents are as follows: 1. 2. 3. 4. 5.
ππ β ππ = ππ+π (ππ )π = ππβπ (ππ)π = ππ β ππ πβπ
π ππ
6.
ππ = ππ π π π π ππ
ππβπ
7. 8.
= ππ = π
=
ππ ππ π
= ππ
Tip 8: Practice 1. If β2 3 β 82 value of n? A. B. C. D. E.
6 7 8 9 10
4
= 24 π , what is the positive
Tip 8: Practice β’ If 43 + 43 + 43 + 43 = 2π , what is the value of n? A. B. C. D. E.
2 4 6 8 10
Tip 8: Practice β’ If m and n are positive and 5π5 πβ3 = 20π3 π what is the value of m in terms of n? A. B.
C.
1 4π 4 π2 4 π3
D. 2π2 E. 4π2
Tip 8: Practice β’ If a and b are positive integers, πβ4 π β1 = 16, πππ π = π2 , which could be true of the value of a? a. b. c. d. e.
0 2 4 8 12
Tip 8: Practice β’ If π β2 Γ 23 = 27 , π€βππ‘ ππ π‘βπ π£πππ’π ππ π? A. 2 B. 4 C. 8
D. E.
1 4 1 8
Tip 8: Practice (cont.) β’ If p and q are positive integers, πβ3 = 2β6 , and πβ2 = 42 , what is the value of ππ ? a. b. c. d. e.
1 2 3 4 5
Tip 9: Geometric Probability β’ Geometric Probability is the probability dealing with the areas of regions instead of the βnumberβ of outcomes. The equation becomes β’ ππππππππππ‘π¦ =
πΉππ£ππππππ π
πππππ π΄πππ ππ πππ‘ππ π
πππππ
Tip 8: Practice (last 1) β’ If a and b are positive integers and π6 π 4 675, what is the value of π + π ? a. b. c. d. e.
3 4 5 7 8
1 2
=