Proportionality
Rational #'s & opps
Expressions & =
Geometry
Circles& Geometry
100

 Given the table: (x: 1,2,3; y: 4,8,12). Is this proportional? If yes, state the constant of proportionality.

yes the constant is 4

100

 Add: -9 + 14.

5

100

 Solve: x+7=19

x=12

100

 Identify the parts in the formula V=(pi)r2h for a right circular cylinder. (Match r and h to parts.)

r is the radius of the circular base; hh is the cylinder's height.

100

Find the circumference of a circle with diameter 10 units. (Leave answer in terms of pi.) EXACT

10pi 

200

 A recipe calls for 2 cups of flour for every 3/4 cup sugar. How many cups of sugar are needed for 8 cups of flour?

For 8 cups flour: sugar = (3/4 ÷ 2) × 8 = (3/8) × 8 = 3 cups.

200

 Multiply and simplify: (-2/3) × (9/4). Show each step

  • Multiply numerators: -2 × 9 = -18.
  • Multiply denominators: 3 × 4 = 12.
  • Simplify: -18/12 = -3/2. Answer: -3/2.
200

 Solve: 4x−5=15

 4x−5=15⇒4x=20⇒x=5.4x−5=15⇒4x=20⇒


x=5

200

 Find the lateral surface area of a cylinder with radius 3 and height 7. (Leave answer in terms of pi.)

42pi

200

 Find the area of a circle with radius 6 cm. (Leave answer in terms of pipi.) EXACT

12pi

300

Write an equation for the proportional relationship where y is miles and x is hours if a car travels at 55 miles per hour. Then find y when x = 3.

 Equation: y=55xy=55x. When x=3x=3, y=165y=165 miles.

300

 Evaluate using order of operations: 3 + 2 × |−5|. Explain why you do the absolute value first.

  • Absolute value first: |−5| = 5.
  • Then multiply: 2 × 5 = 10.
  • Then add: 3 + 10 = 13. Answer: 13. (Explanation: Absolute value is evaluated before multiplication because it is inside an operation that must be simplified first; use the order of operations: parentheses/absolute value, then multiplication, then addition.)
300

Write an equation and solve: “Three times a number decreased by 4 is 23.”

Equation: 3n−4=23⇒3n=27⇒n=9.3n−4=23⇒3n=27⇒n=9

300

Find the total surface area of a right circular cylinder with radius 2 and height 5. (Leave pi.) draw the net. 

28pi

300

 Find the area of a trapezoid with bases 8 cm and 12 cm and height 5 cm.

50cm2

400

A graph shows a line through (0,0) and (4,10). Determine whether the relationship is proportional and find the unit rate.

 Relationship is proportional; unit rate = 2.5.

the line goes through 0,0 the origin and has a constant of 2.5

400

Divide and simplify the fraction: (−6/25) ÷ (3/5). Show the reciprocal step.

  • Division by a fraction = multiply by its reciprocal.
  • Reciprocal of 3/5 is 5/3.
  • Multiply: (−6/25) × (5/3) = (−6×5) / (25×3) = −30 / 75.
  • Simplify: divide numerator and denominator by 15 → −2 / 5. Answer: −2/5.
400

 Determine whether the expressions 2(x+3)2(x+3) and 2x+62x+6 are equivalent. Show reasoning.

2(x+3)=2x+6; yes, they are equivalent (distribute 2 or factor 2)

400

A cylindrical can has radius 4 cm and volume 201.06 cm3. Find the height (use pi approx 3.14). Round to the nearest tenth

h=4.0 cm

400

A composite figure consists of a rectangle 10 by 6 and a semicircle of radius 3 attached to one 6-unit side. Find the total area (leave pi). DRAW IT! KEEP SEMI CIRCLE IN TERMS OF PI- EXACT!!!


Rectangle area = 10×6=60.10×6=60. Semicircle: diameter 6 ⇒ radius 3, area = (1/2)(pi)r2=(1/2)(pi)(32)=(9/2)pi. Total area = 60+(9/2)pi

500

A model uses scale 1 cm : 3 m. On the model, two points are 12 cm apart. What is the actual distance in meters? Show steps.

Actual distance = 12 cm×3 m/cm=36 m.12 cm×3 m/cm=36 m.

500

Multi-step money problem: An item costs $80. First apply a 15% discount. Then add 6% sales tax to the discounted price. What is the final price? Show each step and round to the nearest cent.

  • Original price = $80.
  • Discount = 15% of $80 = 0.15 × 80 = $12.00.
  • Discounted price = 80 − 12 = $68.00.
  • Tax = 6% of discounted price = 0.06 × 68 = $4.08.
  • Final price = 68.00 + 4.08 = $72.08. Answer: $72.08.
500

Write and solve a two-step equation for this context: “Twice a number plus 8 equals 50.” Then interpret the solution in context.

Equation: 2x+8=50⇒2x=42⇒x=21.2x+8=50⇒2x=42⇒x=21. Interpretation: the number is 21.

500

A water tank is a right circular cylinder with height 6 m and radius 2.5 m. Find the volume in cubic meters and then convert to liters (1 cubic meter = 1000 liters). (Leave pi or use pi approx3.14— state which you use.)

v=117.75 m3

convert to liters=117,750 liters

500

A playground design is a composite figure made of a semicircular sandbox attached to one side of a rectangular play area. The rectangle measures 12 m by 8 m. The semicircle has its diameter along the 8 m side of the rectangle.

a) Find the area of the rectangle.
b) Find the area of the semicircle (leave your answer in terms of ππ).
c) Find the total area of the playground (combine the rectangle and semicircle areas). Show all steps and include units.

a) Rectangle area = length × width = 12×8=96 m2.

b) Semicircle: diameter = 8 m ⇒ radius r=82=4 m
Area of full circle = πr2=π(42)=16π.
Area of semicircle = 12×16π=8π m2

c) Total area = rectangle area + semicircle area = 96+8π m2
(If a numeric approximation is needed, use π≈3.14π≈3.14: 8π≈25.12

 so total ≈ 96+25.12=121.12 m2

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