SAT MATH Dr. John Chung

58 Perfect Tips

Designed to help Students get a Perfect Score on the SAT

Tip 1 – Absolute Value

• The absolute value of x, 𝑥 , is regarded as the distance of x from zero.

• How do we convert the general interval into an expression using absolute value?

• Example: 10 ≤ 𝑥 ≤ 30

– Step 1) Find the midpoint: 10+30

2= 20

– Step 2) Find the distance to either point: 20 − 10 = 10

– Step 3) Substitute: 𝑥 −𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 ≤ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

Tip 1 – Absolute Value (cont.)

• If 𝑥 + 3 < 5, what is the value of x?

• If 𝑥 + 3 > 5, what is the value of x?

Tip 1 – Absolute Value (cont.)

• At a bottling company, a computerized machine accepts a bottle only if the number of fluid ounces is greater than or equal to 5

3

7 and less than or equal to 6

4

7. If the machine

accepts a bottle containing 𝑓 fluid ounces, which of the following describes all possible values of 𝑓?

A. 𝑓 − 6 <4

7

B. 𝑓 − 6 <3

7

C. 𝑓 + 6 >4

7

D. 6 − 𝑓 ≤4

7

E. 𝑓 + 6 ≤4

7

Tip 1 – Absolute Value (cont.)

• 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. 𝐸 − 475 < 25 B. 𝐸 + 475 < 25 C. 𝐸 − 500 > 450 D. 475 − 𝐸 = 25 E. 𝐸 − 475 > 25

Tip #2 - Ratio to Similar Figures

• Two polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are in proportion

• If the ratio of the corresponding lengths is a:b, then the ratio of the areas is 𝑎2: 𝑏2 and the ratio of the volumes is 𝑎3: 𝑏3.

Ratio to Similar Figures (cont)

• 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?

• Solution

– The ratio of areas is 25:4

– 25k = 30 or k = 1.2

– Therefore 4k = 4(1.2) = 4.8

Ratio to Similar Figures (cont.)

• 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. 84

B. 92

C. 105

D. 144

E. 147

Tip 3: Combined Range of Two Intervals

Rules

If 5 ≤ 𝐴 ≤ 10 𝑎𝑛𝑑 2 ≤ 𝐵 ≤ 5, i. 7 ≤ 𝐴 + 𝐵 ≤ 15

ii. 10 ≤ 𝐴 × 𝐵 ≤50

iii. 0 ≤ 𝐴 − 𝐵 ≤ 8

iv. 1 ≤𝐴

𝐵≤ 5

Smallest value ≤ Combined Range ≤ Largest Value

Example 1

• Given 2 ≤ 𝑃 ≤ 8 𝑎𝑛𝑑 1 ≤𝑄 ≤ 4. By how much is the

maximum of 𝑃

𝑄 > the

minimum of 𝑃

𝑄 ?

i. Solution: max is 8

1= 8

ii. Min is 2

4=

1

2= .5

iii. 8 − .5 = 7.5

Tip 3: Combined Range of Two Intervals (cont.)

• If −2 < 𝑥 < 4 𝑎𝑛𝑑 − 3 < 𝑦 < 2, what are all possible values of 𝑥 − 𝑦?

A. −4 < 𝑥 − 𝑦 < 2

B. 1 < 𝑥 − 𝑦 < 7

C. 1 < 𝑥 − 𝑦 < 4

D. −4 < 𝑥 − 𝑦 < 7

E. −5 < 𝑥 − 𝑦 < 7

Tip 3: Combined Range of Two Intervals (cont.)

• The value of p is between 1 and 4, and the value of q is between 2 and 6. Which of the following is a possible value of

𝑞

𝑝?

A. Between 1

2 𝑎𝑛𝑑

2

3

B. Between 2

3𝑎𝑛𝑑 2

C. Between 1

2 𝑎𝑛𝑑 6

D. Between 2 𝑎𝑛𝑑 1

2

E. Between 1

2 𝑎𝑛𝑑 1

1

2

Tip 4: Classifying a Group in Two Different Ways

• Organize the information in a table and use a convenient number – 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?

BOYS GIRLS

Seniors 4

9−

4

15=

8

45

2

3∙2

5=

4

15

4/9

Juniors 3

5−

8

45=19

45

5/9

3/5 2/5 1

Tip 4: Classifying a Group in Two Different Ways (cont.)

• 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.

Males Females

> 50 1

5∙ 4 =

4

5= 0.8 2 −

4

5= 1.2

2

≤ 50 3 − 1.2 = 1.8 5

4 3 7

Tip 5: Direct Variation

• When 2 variables are related in such a way that 𝑦 = 𝑘𝑥, the 2 variables are said to be in direct variation.

• Expression of direct variation: i. 𝑦 = 𝑘𝑥

ii.𝑦

𝑥= 𝑘

• Geometric interpretation: 𝑦 = 𝑘𝑥 is a special linear equation where the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 is (0, 0).

• In the 𝑥𝑦 − coordinate plane, 𝑦 = 𝑘𝑥, where 𝑘 𝑖𝑠 𝑠𝑙𝑜𝑝𝑒, but 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 must be zero.

Tip 5: Direct Variation (cont.)

• 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.)

• The 2 adjoining triangles are similar. What is the length of side DF? – 6

–82

13

–90

13

– 8

– 100

13

12

18

5

Tip 6: Inverse Variation

• When 2 variables are related in such a way that 𝑥𝑦 = 𝑘, the two variables are said to be in inverse variation.

– Properties:

• The value of two variables change in an opposite way; that is, as one variable increases, the other decreases.

• The product 𝑘 is unchanged

Tip 7: Special Triangles

• Angle-based Right Triangle – 30-60-90 triangle

• In a triangle whose angles are in the ratio 1:2:3, the sides are in the ratio 1, 3, 2

– 45-45-90 Triangle • In a triangle whose 3 angles are in the ratio 1:1:2, the sides are in

the ratio 1, 1, 2

• Side-based triangles • 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.)

• An equilateral triangle ABC is inscribed inside a circle with radius = 10.

– What is the area of ∆ABC?

B

A C

Tip 7: Special Triangles (cont.)

• Figure ABDE is a square and ∆BCD is an equilateral triangle. If the area of ∆BCD is 16 3 – What is the area of the

square? A. 32

B. 32 3

C. 64

D. 64 2

E. 72

Tip 8: Exponents

• The exponent is the number of times the base is used as a factor.

52 = 25 5 = 𝑏𝑎𝑠𝑒

2 = 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡25 = 𝑝𝑜𝑤𝑒𝑟

• The mathematical operations of exponents are as follows: 1. 𝒂𝒎 ∙ 𝒂𝒏 = 𝒂𝒎+𝒏 2. (𝒂𝒎)𝒏 = 𝒂𝒎∙𝒏 3. (𝒂𝒃)𝒎= 𝒂𝒎 ∙ 𝒃𝒎

4. 𝒂−𝒎 =𝟏

𝒂𝒎

5. 𝒂𝟎 = 𝟏

6.𝒂𝒎

𝒂𝒏= 𝒂𝒎−𝒏

7.𝒂

𝒃

𝒎=

𝒂𝒎

𝒃𝒎

8. 𝒂𝒎𝒏

= 𝒂𝒎

𝒏

Tip 8: Practice

1. If −2 3 ∙ 82 4 = 24 𝑛, what is the positive value of n?

A. 6

B. 7

C. 8

D. 9

E. 10

Tip 8: Practice

• If 43+ 43+ 43+ 43 = 2𝑛, what is the value of n?

A. 2

B. 4

C. 6

D. 8

E. 10

Tip 8: Practice

• If m and n are positive and 5𝑚5𝑛−3 = 20𝑚3𝑛 what is the value of m in terms of n?

A.1

4𝑛

B.4

𝑛2

C.4

𝑛3

D. 2𝑛2

E. 4𝑛2

Tip 8: Practice

• If a and b are positive integers, 𝑎−4𝑏 −1 =16, 𝑎𝑛𝑑 𝑏 = 𝑎2, which could be true of the value of a?

a. 0

b. 2

c. 4

d. 8

e. 12

Tip 8: Practice

• If 𝑘−2 × 23 = 27, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘?

A. 2

B. 4

C. 8

D.1

4

E.1

8

Tip 8: Practice (cont.)

• If p and q are positive integers, 𝑝−3 = 2−6, and 𝑞−2 = 42, what is the value of 𝑝𝑞 ?

a. 1

b. 2

c. 3

d. 4

e. 5

Tip 8: Practice (last 1)

• If a and b are positive integers and 𝑎6𝑏41

2 =675, what is the value of 𝑎 + 𝑏 ?

a. 3

b. 4

c. 5

d. 7

e. 8

Tip 9: Geometric Probability

• Geometric Probability is the probability dealing with the areas of regions instead of the “number” of outcomes. The equation becomes

• 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑅𝑒𝑔𝑖𝑜𝑛

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑔𝑖𝑜𝑛

Tip 9: Geometric Probability

• Geometric Probability is the probability dealing with the areas of region instead of the number of outcomes. The equation becomes

Probability = 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑅𝑒𝑔𝑖𝑜𝑛

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑔𝑖𝑜𝑛

• A typical Problem might be this:

– If you are throwing a dart at the rectangular target below and are equally likely to hit any point on the target, what is the probability that you will hit the small square?

• Probability = 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒

𝑡𝑜𝑡𝑎𝑙=

25

250=

1

10

– This means there is a 1 in 10 chance that a dart thrown at the rectangle will hit the small square

Means 5 cm

25 cm

10 cm

Tip 10: Domain & Range

• The domain of a function is the complete set of input values for which the function is defined.

– The denominator of a fraction cannot be zero

– The number inside a square root sign must be positive

• The range of a function is the set of all output values produced by that function

• If a function 𝑓 is given by 𝑓 𝑥 =𝑥

𝑥−3, which

of the following represents its domain?

A. 𝑥 ≥ 0

B. 𝑥 ≠ 3

C. 𝑥 ≥ 3

D. 𝑥 ≥ 0 𝑎𝑛𝑑 𝑥 ≠ 3

E. All real x

• If a function is given by 𝑔 𝑥 = 𝑥 − 2 − 5, which of the following represents its range?

A. 𝑦 ≥ 0

B. y ≥ 2

C. 𝑦 ≥ 5

D. 𝑦 ≥ −5

E. 𝑦 ≤ −5

Tip 11: Linear Function

• Functions are called “linear” because they form straight

lines in the 𝑥𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑝𝑙𝑎𝑛𝑒. Such a function can be written as:

1) Slope-Intercept form

• 𝑦 = 𝑚𝑥 + 𝑏, where m is the slope and b is the y intercept

2) Point – Slope form

• 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 , where 𝑥1, 𝑦1 is the known point on the line

3) General form: 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

4) Standard form: 𝐴𝑥 + 𝐵𝑦 = 𝐶

Tip 11 Practice

• For a linear function 𝑓, 𝑓 0 = 2 𝑎𝑛𝑑 𝑓 3 =5. 𝐼𝑓 𝑘 = 𝑓(5), 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘?

A. 5

B. 6

C. 7

D. 8

E. 9

• The adjoining table shows some values for the function 𝑓. If 𝑓 is a linear function, what is the value of a + b?

A. 24

B. 36

C. 48

D. 60

E. Cannot be determined

𝑥 𝑓(𝑥)

0 𝑎

1 12

2 𝑏

• A linear function is given by 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and 𝑎 > 0, 𝑏 < 0, 𝑎𝑛𝑑 𝑐 > 0. Which of the following graphs best represents the graph of the function?...

• If 𝑓 is a linear function and 𝑓 3 = 2 and 𝑓 5 = 6, what is the y-intercept of the graph of 𝑓?

A. 4

B. 2

C. 0

D. -2

E. -4

• If 𝑓 is a linear function and 𝑓 3 = −2 and 𝑓 4 = −4, what is the x-intercept of the graph of 𝑓?

A. 3

B. 2.5

C. 2

D. 0

E. -1

• The adjoining figure shows the graph of function 𝑓. If b = 2a, what is the value of a? A. 2

B.5

2

C.15

13

D.5

4

E.3

2

t h(t)

-1 6

0 4

1 2

2 0

• The table shows some values for the linear function h for selected values of t. Which of the following defines h?

A. ℎ 𝑡 = 4 − 𝑡

B. ℎ 𝑡 = 4 − 2𝑡

C. ℎ 𝑡 = 4 + 2𝑡

D. ℎ 𝑡 = 4 + 𝑡

E. ℎ 𝑡 = 2 − 0.5𝑡

• Fahrenheit (F) and Celsius are related by

𝐹 =9

5𝐶 + 32. If the Fahrenheit temperature

is increased by 27 degrees, what is the degree increase in Celsius temperature? A. 15

B. 20

C. 32

D. 59

E. 81

• In the formula 𝑃7

12𝑄 + 60, 𝑖𝑓 𝑃 is increased

by 35, then what is the increase in Q ?

A. 35

B. 60

C. 80

D. 140

E. 160

Warm-up

• In the figure, a circle is tangent to line l, x-axis, and y-axis. If the radius of the circle is 5, what is the value of t?

A. 7

B. 8

C. 9

D. 10

E. 11

Tip 12: Triangle Inequality

• Theorem 1 – The length of one side of a triangle is less than the sum of the other 2 sides and is greater than the difference of the other 2 sides.

• Theorem 2 – The longest side has the largest opposite angle

• Theorem 3 – The measure of an exterior angle is equal to the sum of its two non-adjacent interior angles

• The lengths of the sides of a triangle are 3, x + 3, and 9. Which could be the value of x ?

A. 1

B. 2

C. 3

D. 4

E. 9

• Which of the following cannot be possible to construct a triangle with the given side lengths?

A. 6, 7, 11

B. 3, 6, 9

C. 28, 34, 39

D. 35, 120, 125

E. 40, 50,60