Evaluating Algebraic Expressions
Sarah buys 3 notebooks that cost $x dollars each and 2 pens that cost $y dollars each. If notebooks cost $4 each and pens cost $2 each, what is the total cost?
What is $16?
(3x + 2y = 3(4) + 2(2) = 12 + 4 = 16)
Maria has $15 more than twice what Jake has. If Maria has $47, how much does Jake have?
What is $16?
(2x + 15 = 47, so 2x = 32, therefore x = 16)
The formula for the area of a triangle is A = ½bh. Solve for the height h when the area is 24 square feet and the base is 8 feet.
What is 6 feet? (24 = ½(8)h, so 24 = 4h, therefore h = 6)
A taxi charges $3 for the first mile and $2 for each additional mile. Write the linear function for the total cost C in terms of miles m (where m ≥ 1), then find the cost for a 5-mile trip.
What is $11?
(C = 3 + 2(m-1) = 2m + 1, so for m = 5: C = 2(5) + 1 = 11)
Two numbers have a sum of 25 and a difference of 7. Find the larger number.
What is 16?
(x + y = 25 and x - y = 7. Adding: 2x = 32, so x = 16)
The perimeter of a rectangle is given by P = 2l + 2w. Find the perimeter when the length is 8 feet and the width is 5 feet.
What is 26 feet?
(P = 2(8) + 2(5) = 16 + 10 = 26)
A pizza costs $12 plus $1.50 for each topping. If Tom pays $19.50 for his pizza, how many toppings did he get?
What is 5 toppings?
(12 + 1.5x = 19.5, so 1.5x = 7.5, therefore x = 5)
he formula for simple interest is I = prt. If you earn $120 interest on a principal of $2000 over 2 years, what is the interest rate?
What is 3% or 0.03? (120 = 2000(r)(2), so 120 = 4000r, therefore r = 0.03 = 3%)
The water level in a pool decreases at a constant rate. After 2 hours, there are 800 gallons, and after 5 hours, there are 650 gallons. How many gallons were there initially?
What is 900 gallons?
(Rate = (650-800)/(5-2) = -50 gal/hr. Using point-slope: y - 800 = -50(x - 2), so y = -50x + 900. Initially (x = 0): y = 900)
Adult tickets cost $8 and child tickets cost $5. If 120 tickets were sold for a total of $810, how many adult tickets were sold?
What is 70 adult tickets?
(a + c = 120 and 8a + 5c = 810. From first equation: c = 120 - a. Substituting: 8a + 5(120 - a) = 810, so 3a = 210, therefore a = 70)
A car rental company charges $25 per day plus $0.15 per mile. If the expression for total cost is 25d + 0.15m, find the cost for renting a car for 3 days and driving 200 miles.
What is $105?
(25(3) + 0.15(200) = 75 + 30 = 105)
The sum of three consecutive integers is 84. What is the smallest of these integers?
What is 27?
(x + (x+1) + (x+2) = 84, so 3x + 3 = 84, 3x = 81, x = 27)
The temperature conversion formula is F = (9/5)C + 32. What Celsius temperature equals 86°F?
What is 30°C? (86 = (9/5)C + 32, so 54 = (9/5)C, therefore C = 54 × 5/9 = 30)
A phone plan costs $30 per month plus $0.10 per text message. If someone's monthly bill is $45, how many text messages did they send?
What is 150 text messages?
(45 = 30 + 0.10x, so 0.10x = 15, therefore x = 150)
A coffee shop sells regular coffee for $2 per cup and premium coffee for $3 per cup. Yesterday they sold 85 cups total and made $230. How many cups of regular coffee did they sell?
What is 40 cups?
(r + p = 85 and 2r + 3p = 230. From first: p = 85 - r. Substituting: 2r + 3(85 - r) = 230, so -r = -25, therefore r = 40)
The surface area of a cylinder is given by SA = 2πr² + 2πrh. Find the surface area when r = 3 inches and h = 8 inches. (Use π ≈ 3.14)
What is approximately 207.24 square inches?
(SA = 2(3.14)(3)² + 2(3.14)(3)(8) = 2(3.14)(9) + 2(3.14)(24) = 56.52 + 150.72 = 207.24)
A gym membership costs $40 to join plus $25 per month. If Sarah wants to spend no more than $190, what is the maximum number of months she can have the membership?
What is 6 months?
(40 + 25x ≤ 190, so 25x ≤ 150, therefore x ≤ 6)
The formula for the volume of a cone is V = (1/3)πr²h. If a cone has a volume of 84π cubic inches and a radius of 6 inches, find its height.
What is 7 inches?
(84π = (1/3)π(6)²h, so 84π = 12πh, therefore h = 7)
The relationship between Celsius and Fahrenheit temperatures is linear. Water freezes at 0°C (32°F) and boils at 100°C (212°F). Find the Fahrenheit temperature when it's 25°C.
What is 77°F?
(Slope = (212-32)/(100-0) = 1.8. Using point-slope: F - 32 = 1.8(C - 0), so F = 1.8C + 32. For C = 25: F = 1.8(25) + 32 = 77)
The perimeter of a rectangle is 44 cm, and its length is 6 cm more than its width. Find the dimensions of the rectangle.
What is width = 8 cm and length = 14 cm?
(2l + 2w = 44 and l = w + 6. Substituting: 2(w + 6) + 2w = 44, so 4w = 32, therefore w = 8 and l = 14)
A company's profit is modeled by P = -2x² + 50x - 200, where x is the number of items sold in hundreds. Find the profit when 1,200 items are sold (x = 12).
What is $112?
(P = -2(12)² + 50(12) - 200 = -2(144) + 600 - 200 = -288 + 600 - 200 = 112)
The length of a rectangle is 3 less than twice its width. If the perimeter is 54 meters, find the width of the rectangle.
What is 10 meters? (l = 2w - 3, and 2l + 2w = 54, so 2(2w - 3) + 2w = 54, which gives 6w - 6 = 54, therefore w = 10)
The lens equation in physics is 1/f = 1/s + 1/s', where f is focal length, s is object distance, and s' is image distance. If f = 12 cm and s = 20 cm, find s'.
What is 30 cm?
(1/12 = 1/20 + 1/s', so 1/s' = 1/12 - 1/20 = 5/60 - 3/60 = 2/60 = 1/30, therefore s' = 30)
A company's revenue R (in thousands) is related to advertising spending A (in thousands) by R = 50 + 3A. Their profit P equals revenue minus the sum of advertising costs and $80,000 in other costs. If they want a profit of $40,000, how much should they spend on advertising?
What is $35,000?
(P = R - A - 80, and R = 50 + 3A, so P = 50 + 3A - A - 80 = 2A - 30. Setting P = 40: 40 = 2A - 30, so A = 35, which is $35,000
A boat travels 60 miles downstream in 2 hours and takes 3 hours to travel the same distance upstream. Find the speed of the boat in still water and the speed of the current.
What is boat speed = 25 mph and current speed = 5 mph?
(Let b = boat speed, c = current speed. Downstream: b + c = 60/2 = 30. Upstream: b - c = 60/3 = 20. Adding equations: 2b = 50, so b = 25. Then c = 30 - 25 = 5)