[PSAT] 5-MONTH Revision

5-Month Revision

​Percent, Ratio & Rate, Function notation, Linear relationships

Percent

Converting from percent to number and from number to percent

Percent to number: Divide by 100 (75% = 0.75).

Number to percent: Multiply by 100 (0.4 = 40%).

Multiple Choice

125% as decimal?

• 0.125

• 1.25

• 12.5

• 1.025

Multiple Choice

0.08 to percent?

• 0.8

• 80

• 88

• 8

Increasing and decreasing by percent

​Increase: Multiply by 1 + (percent ÷ 100) (e.g., 20% increase: × 1.2).

Decrease: Multiply by 1 - (percent ÷ 100) (e.g., 15% decrease: × 0.85).

Example: A $200 phone increases by 10%. New price: 200 × 1.1 = 220.

Multiple Choice

A $40 book decreases by 20%. What’s the new price?

• 32

• 36

• 28

• 30

Multiple Choice

A $50 item increases by 30%. What’s the new price?

• 60

• 65

• 70

• 75

Cross-multiplication rule: what comes after than,from will be 100%

Multiple Choice

15 is what percent of 60?

• 20

• 25

• 30

• 15

Multiple Choice

40 is 80% of what number?

• 32

• 48

• 50

• 60

Percent change formula/questions, identifying which one is old or new.

​Percent change = (new - old) / old × 100%.
Identify old and new values carefully from context.

Example: Price rises from $100 to $120.

Percent change: (120 - 100) / 100 × 100% = 20%.

Multiple Choice

A shirt’s price drops from $25 to $20. What’s the percent decrease?

• 20

• 25

• 15

• 30

Multiple Choice

A stock rises from $50 to $65. What’s the percent increase?

• 15

• 20

• 35

• 30

Solving percent word problems

​Explanation:
1) Percentages measure parts of a whole, used in discounts, taxes, grades, and more.
For example, a 20% sale means you pay 80% of the original price.

Example: A $50 shirt is on a 30% discount.

Savings: 30% × 50 = 0.3 × 50 = 15.

Sale price: 50 - 15 = 35.
2) Translate word problems into equations using percent conversions or cross-multiplication. Identify key quantities (part, whole, percent).

Multiple Choice

A store offers a 25% discount on a $80 jacket. What’s the sale price?

• 20

• 60

• 50

• 65

Multiple Choice

A $120 TV has a 15% tax. What’s the total cost?

• 135

• 138

• 140

• 150

Ratio & Rate

Basic unit conversion(without square)

Convert measurements using a conversion factor.

Formula: Converted Value = Original Value × Conversion Factor

Multiple Choice

If 1 mile = 1.609 kilometers, how many kilometers are in 5 miles?

• 3.106 km

• 8.045 km

• 5 km

• 8 km

Multiple Choice

If 1 gallon = 3.785 liters, how many liters are in 2 gallons?

• 7.57 liters

• 0.528 liters

• 2 liters

• 7.57 liters

Direct and indirect relationship

​Direct Relationship: When one value increases, the other does too.

Example: Months worked and money earned.
If you earn $100 per month, 6 months = (100 × 6 = 600).

Formula: y = kx

Indirect Relationship: When one value increases, the other decreases.

Example: Speed and time for a fixed distance. At 50 mph, 100 miles takes 2 hours; 150 miles takes (150 ÷ 50 = 3) hours.

Formula: y = k ÷ x

Multiple Choice

If you earn $100 per month, how much total in 6 months?

• 100

• 60

• 600

• 500

Multiple Choice

A car travels 100 miles in 2 hours. Time for 150 miles at the same speed?

• 3 hours

• 1.5 hours

• 2 hours

• 4 hours

Ratio Rate word problems

​Ratios compare two quantities showing their relative sizes.
For example, in a class of 35 students with 20 girls and 15 boys,
the ratio of girls to boys is 20 to 15, which simplifies to 4 to 3.

Multiple Choice

A recipe requires 6 cups of flour and 3 cups of sugar. How much flour do you need for one-third of the recipe?

• 2 cups

• 3 cups

• 1 cup

• 4 cups

Multiple Choice

Two workers together can paint a fence in 5 hours. How long will it take one worker to paint the fence alone?

• 2.5 hours

• 5 hours

• 10 hours

• 15 hours

Function Notation

Input and output concepts

Input (Domain): The value you put into a function, usually written as x.

Output (Range): The result you get from the function, usually written as f(x).

Example: For f(x) = x², input 3 gives output 9.

Fill in the Blanks

For f(x) = 3x - 1, what is the output when the input is 4?

Fill in the Blanks

For $g(t)=\frac{t}{2}+3,$ what input gives an output of 5

One input cannot produce 2 outputs

One Input Cannot Produce Two Outputs

A function must have exactly one output for each input.

Example: f(x) = x + 2 gives output 5 for input 3, not 5 and 6.

Multiple Choice

Which set of pairs is a function?

A) { (1,2) , (1,3) }

B) { (2,3) , (3,4) }

• only A

• only B

• Both A and B

• Neither A nor B

Multiple Choice

Is { (0,0) , (1,1) , (1,-1) } a function?

• Yes

• No

• Can’t be determined

Plugging in

​Plugging in means substituting a value into a function to find the output.

Example: For f(x) = 2x + 1, f(3) = 2(3) + 1 = 7

Fill in the Blanks

For h(x) = x² - 3x, what is h(-2)?

Fill in the Blanks

For $k(a)=\frac{a+1}{a-1}$ , what is k(2)?

Composite function notations

​A composite function combines two functions:

(f ∘ g)(x) = f(g(x)) — you plug g(x) into f(x)

Example:

If f(x) = x + 3 and g(x) = x², then f(g(x)) = x² + 3

Fill in the Blanks

If f(x) = 2x and g(x) = x - 1, what is f(g(3))?

Fill in the Blanks

If h(x) = x² and k(x) = 3x + 1, what is h(k(1))?

X and Y intercepts

​X-Intercept: Where the graph crosses the x-axis → y = 0

Y-Intercept: Where the graph crosses the y-axis → x = 0

Example: For f(x) = 2x - 6:

X-intercept: 0 = 2x - 6 → x = 3 → point (3, 0)

Y-intercept: f(0) = 2(0) - 6 = -6 → point (0, -6)

Fill in the Blanks

What is the x-intercept of f(x) = 3x - 6?

Multiple Choice

What is the y-intercept of g(x) = x² - 5?

• (5,0)

• (0,-5)

• (0,5)

• (-5,0)

Real-life application of function

​Functions describe how one quantity depends on another
in real-world contexts.

Example:

1) Taxi Cost Function:

C(d) = 2.5 + 1.5d

This gives the total taxi fare C based on the distance d in miles.

There’s a base fare of $2.50 and a $1.50 charge per mile.

Fill in the Blanks

What Celsius value gives 32°F in the function F(C) = (9/5)C + 32?

Fill in the Blanks

If the taxi fare is $7.00 and the base fare is $2.50, how many miles were traveled?

Use: C(d) = 2.5 + 1.5d

Linear Relationships

Interpretation of rate of change and its application in contexts real-life

applications

​Linear relationships show a steady, constant change between two things.

For example, if a population grows by a fixed number of people each year, that’s a constant change.
In real life, this could be like a job paying a set amount per hour or a phone plan charging a fixed amount per text.

Multiple Choice

A city had a population of 3,000 in 2015 and 3,400 in 2017. If the population changes by a constant amount each year, what will it be in 2019?

• 3,600

• 3,800

• 4,000

• 3,400

Multiple Choice

A delivery service charges a $5 flat fee plus $1.50 for each mile driven. Write the equation for the total cost based on miles driven, and find the cost for a 4-mile delivery.

• $9.50

• $11.00

• $12.50

• $13.00

Interpretation of x and y intercepts in contexts

​Y-Intercept: This is the value when the independent variable (usually time or distance) is zero.

👉 It tells you the starting amount.

Example: A car rental costs $50 even if you don’t drive anywhere. That $50 is the y-intercept.

X-Intercept: This is when the total or final value becomes zero.

👉 It tells you when something runs out or ends.

Example: A pool starts with 1000 gallons and loses 50 gallons each hour. The x-intercept tells you how many hours until the pool is empty.

Multiple Choice

A streaming service charges a one-time signup fee of $10 and then $5 per month. The total cost c after m months is given by the equation c = 10 + 5m. What does the y-intercept represent in this context?

• The monthly cost

• The total cost after one month

• The signup fee

• The number of months needed to reach a total cost of $10

Multiple Choice

A pool contains 200 gallons of water and is being drained at 25 gallons per minute. The amount of water w left after t minutes is given by w = 200 − 25t. What does the x-intercept represent in this context?

• The time when the pool is empty

• The initial amount of water

• The rate at which the pool is being drained

• The amount of water drained per minute

Determining the function by given points

​If you're given two points, you can find a linear function (an equation like y=mx + b)

by calculating the slope and the y-intercept.

STEPS:

1. Find the Slope (m):

The slope is the constant rate of change, calculated as the change in y divided by

the change in x between the two points:

m = (y₂ - y₁)/(x₂ - x₁)

2. Find the Y-Intercept (b):

Use the slope and one point in the equation y = mx + b to solve for b.

3. Write the Equation:

Combine m and b into y = mx + b.

Multiple Choice

A line passes through the points (2, 2) and (4, 6). What is the equation of the line?

• y = 2x + 0

• y = 2x − 2

• y = 3/2 x + 1

• y = 3/2 x − 1

Multiple Choice

A line passes through the points (0, 5) and (4, 1). What is the equation of the line?

• y = −x + 5

• y = −x + 1

• y = −1/4 x + 5

• y = −1/4 x + 1

Generating the function by the given context

​To create a linear function from a real-world situation, identify the starting value

(y-intercept) and the constant rate of change (slope), then form the equation.

STEPS:

1. Identify the Initial Value:

This is the value of the dependent variable when the independent variable is zero.

(What you start with before any change happens)

2. Determine the Rate of Change:

This is how much the dependent variable changes for each unit increase in the

independent variable. (How much it increases/decreases per unit)

3. Form the Equation:

Use the format: y = initial value + slope × x

Or in standard form: y = mx + b

Multiple Choice

A student starts with $50 in a savings account and deposits $20 each month. What is the equation that represents the amount a in the account after m months?

• a = 50 + 20m

• a = 20 + 50m

• a = 50m + 20

• a = 20m − 50

Multiple Choice

A car rental company charges a flat fee of $30 plus $0.25 for each mile driven. What is the equation for the total cost c for driving d miles?

• c = 30 + 0.25d

• c = 0.25 + 30d

• c = 30d + 0.25

• c = 0.25d − 30

4-Month Revision

Quadratic equations, Absolute value, Advanced system of equations

Quadratic equations

- Real-life application: Quadratic equations model real-world scenarios like calculating area or tracking the height of a projectile.

- Discriminant (D = b² − 4ac): Tells you the type of roots a quadratic has. If D> 0, there are 2 real roots; if D = 0, 1 real root; if D < 0, no real roots.

- Formula for roots: The quadratic formula x = (−b ± √D) / (2a) helps solve any quadratic equation of the form ax² + bx + c = 0.

- Vieta's theorem: For ax² + bx + c = 0, the sum of the roots is −b/an and the product is c/a.

- Number of solutions: The number of real solutions is determined by the value of the discriminant.

- Factored form: Any quadratic can be expressed as a(x − x₁)(x − x₂) = 0, where r₁ and r₂ are the roots.

- Biquadratic equations: Equations like ax⁴ + bx² + c = 0 can be simplified by letting y = x², reducing them to quadratic form.

- Exponential to quadratic: If you see a square root (like √(expression)) equal to something else, square both sides to remove the root.

Multiple Choice

A rectangle’s length is 3 m more than its width, and its area is 40 m². What’s the perimeter?

• 20 m
• 24 m
• 26 m
• 30 m

Multiple Choice

For 2x² + 3x − 5 = 0, calculate Discriminant and find number of the roots.

• D=0

  One real root

• D=-49

  No real roots

• D=36;

  Three distinct real roots

• D=49;

  Two distinct real roots

Multiple Choice

Find the zeros of:

4x² − 4x - 3 = 0

using Discriminant

• x = 2, x = -1
• x = 0, x = -3
• x = 1, x = -1.5
• x = 1.5, x = -0.5

Multiple Choice

For x² + 5x + 6 = 0, find the sum and product of the roots.

• Sum: -3, Product: 8
• Sum: -4, Product: 7
• Sum: -5, Product: 6
• Sum: -6, Product: 5

Multiple Choice

How many real solutions does 4x² + 4x + 1 = 0 have?

• 2
• 0
• 3
• 1

Multiple Choice

Write this equation in factored form

x² − 6x + 8 = 0

• (x - 1)(x - 8)
• (x - 2)(x - 4)
• (x + 2)(x + 4)
• (x + 6)(x - 2)

Multiple Choice

Solve x⁴ − 5x² + 4 = 0.

• x = 0
• x = ±3
• x = ±1, ±2
• x = ±4

Multiple Choice

Solve the equation: $$\sqrt[]{x+3}=5$$

• 10
• 15
• 18
• 22

Multiple Choice

Two consecutive integers have a product of 72. Find them.

• 8 and 9
• 6 and 7
• 9 and 10
• 7 and 8

Absolute Value

- Real-life application: Absolute value measures distance or magnitude, always as a positive number, regardless of direction.

- Solving equations: Equations like |x − a| = b have two solutions: x = a + b or x = a − b.

- Solving inequalities:

- |x| < a means −a < x < a

- |x| > a means x < −a or x > a

- Word problems: Absolute value is used to express acceptable ranges or limits. For example, temperatures staying within 5°C of 25°C are written as |t − 25| ≤ 5.

Multiple Choice

If you walk 5 km to school and back, what’s the total distance walked?

• 10 km
• 15 km
• 20 km
• 5 km

Multiple Choice

Solve |x − 3| = 7.

• x = 3
• x = 5
• x = 10, x = -4
• x = -10

Multiple Choice

Solve |2x + 1| ≤ 5.

• [-3, 2]
• [-2, 5]
• [0, 3]
• [-4, 1]

Multiple Choice

Write an absolute value inequality for temperatures between 20°C and 30°C.

• |x - 25| > 5
• |x - 30| < 5
• |x - 20| < 10
• |x - 25| < 5

Advanced system of equations

- Advanced system of equations = Normal system of equations + Exponential equation.

These systems mix linear and nonlinear parts.
Use substitution, factoring, or identity tricks to solve.

Pattern Use:

Use identities like x³ + y³ = (x + y)(x² - xy + y²)

Tips:

- Use substitution if isolation is easy

- Try test values like 0, 1, or -1

- Watch for quadratics or factorable forms

Multiple Choice

Solve the given system of equations:
4x + 5y = 14

y² − x² = 3

• (-121/9 , -122/9 ) and (1

• (-121/9, 122/9 ) and (-1,-2)

• (122/9, -121/9) and (-1,2)

• (-121/9, 122/9 ) and (1,2)

Multiple Choice

Find all possible solutions in this equation :
x + y = 5

x³ + y³ = 125

• (25,0) and (0,25)

• only (5,0)

• only (0, 5)

• (0, 5) and (5, 0)