College Math Prep A - Unit 1

Tiana is building her dog a rectangular kennel that is 29 feet long, and she purchased 108 feet of fencing. She needs to know how wide the kennel should be to use all the fencing.

Tiana can draw a rectangle and label two opposite sides 29 feet long, then subtract those two side values from 108 and divide the difference by 22 to find the width of the kennel.

Katie's parents rented a banquet room for her Sweet 16 party. The cost of the room was $250 plus $39 per guest who attended. What equation can be used to find the total cost of renting the room for the party?

c = 39g+250, where c is the total cost and g is the number of guests.

Sam, Joe, Caleb, and Mike, are going to rent an apartment together. Since the bedrooms have various sizes, the friends have agreed to pay different portions of the rent. Sam will pay 1/2 of the monthly rent, Joe will pay 1/4, Caleb will pay 1/8, and Mike will pay the rest. If the rent is $2,400 per month, how much will Mike pay?

Find the fraction of the rent Mike has to pay and multiply that fraction by $2,400.

Mobile Provider A charges a $35 monthly fee plus $0.08 per minute for calls. Mobile Provider B charges a $25 monthly fee plus $0.14 per minute. What is an appropriate process to find when the two plans cost the same?

Make a graph with the number of minutes on the x-axis and the cost on the y-axis. For Provider A, start at 0 minutes and $35, and increase the cost $0.08 for every minute. For Provider B, start at $25, and increase the cost $0.14 for every minute. Continue both patterns until the plans have the same cost.

Declan earned $48 in tips last night working as a waiter. He spent $12 on new guitar strings and half the remaining money at the movies. How much does he have left? In order to find out, Declan added $48 and $12, then took half the result to find that he had $30 left. He is incorrect. What is an appropriate problem-solving strategy that gives the correct solution?

Find the difference between $48 and $12, then take half the result.

Leah hangs 6 posters in a row, each 14 inches wide, on a bedroom wall. There are 4 inches between each poster. For the posters at each end, there are 3 inches between each poster and the end of the wall.

Leah can draw a picture showing each poster and space. Then she can add the measurements together to find the width of the wall.

The sum of two angle measures is 110 degrees. The larger angle is 10 degrees less than twice the smaller angle. What is a process for finding the measure of the smaller angle?

Let n represent the measure of the smaller angle. Then 10 less than twice the smaller angle is 2n−10, so the sum of the angles is 2n−10+n=110. Solve the equation using inverse operations to find the measure of the smaller angle.

Dimitria knows the difference of two angle measures is 130 degrees. The larger angle is 40 degrees more than 4 times the smaller angle. She wants to find the measure of the smaller angle. Dimitria takes the difference between 130 and 40 to get 90, then multiplies by 4 to get 360 degrees for the smaller angle. That is incorrect. Which process solves the problem correctly?

Let x represent the measure of the smaller angle. Then 40 more than 4 times the smaller angle is 4x+40, so the difference of the angles is 4x+40−x=130. Solve the equation using inverse operations to find the measure of the smaller angle.

A tip jar at a coffee shop contains three times as many quarters as dollar bills. It has $8.75 in total, and Zach wants to determine the number of quarters and dollar bills in the jar. Zach makes two tape diagrams as shown, one with 3 rectangles and one with 1 rectangle. He labels the diagrams with "$8.75.{}^{\prime\prime} Zach divided 8.75 by 4 to get 2.1875. He rounded down, wrote "2{}^{\prime\prime} on each rectangle, and stated that 6 quarters and 2 dollar bills are in the tip jar. That is incorrect. Which strategy correctly solves the problem?

Make a table with a column for "dollars," a column for "quarters," and a column for "value." Start with 1 dollar and 3 quarters, and calculate the value. Continue increasing the dollar column by 1 and the quarter column by 3 until the value reaches $8.75

A taxi from the airport charges a base fee of $5.25 plus $0.60 per mile. The subway costs $2.00 for a transit card plus $1.00 per stop (the stops are each 11 mile apart). What is an appropriate process for calculating when it becomes cheaper to take a cab?

Make a table with columns for "Miles," "Cab Cost," and "Subway Cost." For each mile, calculate the cab cost by adding another $0.60 to $5.25. Then calculate the subway cost by adding another $1.00 to $2.00. Continue to increase the miles used in the calculations until the cost of the subway is greater than that of the cab.

Malik is cashing his $140 paycheck and would like only $5 and $10 bills. The bank teller gives him 18 bills. What is an appropriate process for finding how many $5 bills he received?

Make a table with three columns, labeled "$5 bills," "$10 bills," and "Total Value." Start with 0 $5 bills, 18 $10 bills, and a total value of $180. Then increase the $5 bills by 1 and decrease the $10 bills by 1 to find the combination of bills that adds up to $140.

Taylor's baseball team played a full 9-inning game. The team scored 4 runs in every inning, and for every 2 runs it scored, it had 5 hits. What is an appropriate process for finding how many total hits the team had?

Make a tree diagram with 1 inning as the starting point. Draw 4 branches to represent the runs. Connect every 2 runs to a single node and draw 5 branches to represent the hits. Multiply the number of hits by 99 to find the number of hits in the full 9-inning game.

Maria's parents are renting a banquet hall for her Quinceanera. The hall costs $425 plus $25 per guest. If 100 guests attend the party, what will be the total cost to rent the hall? After analyzing the problem, what is an appropriate process for solving it?

Maria's parents will pay a base charge of $425 plus $25 for each of the 100 guests. Find 425+25(100) for the total amount they will pay to rent the hall.

A taxi company charges a pickup fee of $5.25 plus $0.60 per mile. What is an appropriate process for finding the cost of a 7-mile ride?

The initial value is $5.25, and the rate of change is $0.60. So, write the equation c=0.60m+5.25, where c is the cost and m is the number of miles. Substitute 7 for m and simplify the right side of the equation to get the cost.

Jaime purchased a sewing machine. After 8% sales tax, she paid $139.32. What equation can be used to find the pre-tax price of the sewing machine?

p+0.08p=139.32, where pp is the pre-tax price of the machine.

Phil and Ava both worked 14 hours this week. Phil earned $112.70, and Ava earned $113.82. Ava wants to know how much more per hour she earns than Phil. Which strategy is appropriate for Ava to use?

Ava can divide each of their weekly earnings by 14,14, then subtract the results to find out how much more she earns per hour than Phil.

A piece of quartz has a mass of 13.5 g and a volume of 10.24 cm3. Using context clues from the problem, what formula can be used to find the density of the quartz?

D=m/V

Dana's parents are hosting a graduation party for her. Her favorite restaurant charges $150 to rent the party room plus $16 per person for food and drinks. How can Dana use the given information to determine how much her parents paid for 20 people to attend her party?

Dana can multiply $16 by 20 and then add $150 for a total of $470.

Julie is sewing a cube-shaped pillow with edges that are 14 inches long. She would like to sew a piece of lace along each of the edges. What model (and related actions) can she use to help her find out how many 14-inch pieces of lace she will need?

Draw a cube and count the number of edges.

Max has a coupon for a free pizza. He gets to pick one item from each category: crust (thin or thick), meat (pepperoni or sausage), and topping (peppers, mushrooms, or olives). What is an appropriate model for Max to use to find how many combinations of pizza he has to choose from?

He could use a tree diagram with the types of crust as starting points, with branches for the types of meats, and a second set of branches for the toppings.

Jake is renting a fishing boat. The marina charges a flat fee of $150 plus $20 per hour to rent the boat. Jake wants to know how much it will cost him to rent the boat for 4 hours.

Jake can multiply $20 by 4, then add $150 to calculate the total cost.

Ellis took a summer job selling chef's knives and is offered two options for how he will be paid. Option 1 pays a base of $40 a day plus $4 per knife sold. Option 2 pays a base of $60 a day plus $2 per knife sold. Which option would pay more if Ellis sells an average of 12 knives a day? How do you know? Select the option that correctly answers both questions.

Option 1, because selling 12 knives under Option 1 would pay $88 a day, while Option 2 would pay $84 a day

Phil is shopping for a new cable plan. Plan 1 costs $50 per month plus $5 per movie watched on demand. Plan 2 costs $56 per month plus $2.50 per movie watched on demand. Phil watches an average of 3 movies on demand per month. If he bases his decision only on price, which cable plan should he choose?

Plan 2, because it will cost $63.50 per month with 3 on-demand movies

Ryen is joining a gym and has two options. Planet Muscle charges a $200 membership fee and $10 per month in dues. NY Fitness does not charge a membership fee but charges $30 per month in dues. If Ryen plans to keep his membership for a year, which gym is the better deal? How do you know? Select the option that correctly answers both questions.

Planet Muscle, because it will cost $320 for a year, while NY Fitness will cost $360

Seth places a 10-foot ladder against his house to reach a window that is 8 feet high. Use the following drawing of this situation to help determine how far the ladder base is from the house. After analyzing the problem, what is an appropriate process for solving it?

Seth knows one triangle leg is 8 feet long, and the hypotenuse is 10 feet long. He needs to find the remaining leg length. Substitute a=8 and c=10 into a2+b2=c2 to find the remaining leg length. Check your answer by making sure the sum of the legs' squares equals the hypotenuse's square.

Jayson is comparing two hotels because he plans to attend a conference. He will need breakfast every morning, a parking spot for his car, and Wi-Fi service. The Ocean Inn charges $500 a week (7 days/6 nights) plus $4.95 per night for Wi-Fi and $10 per night for parking. It provides a free breakfast each day. The Seagull Resort charges $495 a week (7 days/6 nights) plus a $15 resort fee per night, which includes Wi-Fi, parking, and a continental breakfast each day. Based solely on price, which hotel should Jayson choose?

The Seagull Resort, because it costs $4.70 less for the week

A calculator originally priced at $105.99 was discounted to $89.99. What is a process for solving for the percentage of the discount?

The discount price is the difference between the original price and a percentage of the original price. This is modeled by the equation 105.99−(105.99)d=89.99, where d is the percentage of the discount. Solve the equation using inverse operations and rewrite the answer as a percentage.

Andres sells personalized hats at a kiosk in the mall. He earns a base salary of $250 a week plus a commission of $3 per hat sold. How much would he earn for a week in which he sells 60 hats? Examine the following steps used to solve the problem. The solution contains an error. What is an appropriate problem-solving strategy that gives the correct solution?

Use the equation p=3h+250, where pp is total pay and h is the number of hats. Then substitute h=60 to get the value of p.

The sum of two numbers is 25. The difference of the numbers is 11. Find the numbers. After analyzing the problem, what is a good plan for solving it?

Use trial and error by guessing a set of two numbers and checking their sum and difference.

Auggie started with an even number. He added 10, divided that result by 2, then subtracted 5, which gave him 28. After analyzing the problem, what is a good plan to find Auggie's original number?

Work backward: take 28+5, which is 33; multiply that by 2, which is 66; then subtract 10, which is 56. So, Auggie's original number was 56.

Mr. Adams wants to rent a beachfront condominium. Unit 1 costs $1,200 for the week plus a cleaning fee of$50 per day. Unit 2 costs $1,450 for the week with no cleaning fee. What is an appropriate model to use to determine at which number of days both units cost the same amount for the week?

a graph with the number of days for the week on the x-axis and the cost on the y-axis, showing the cost of Unit 1 starting at $1,200 and increasing $50 each day, and the cost of Unit 2 at $1,450 each day without increasing

Which mathematical model is used most often in conjunction with other models?

a table

The new electric Ford Mustang can go from 0 to 60 mph in 3 seconds. Using context clues from the problem, what formula can be used to find the acceleration of the Mustang?

a=v−v0/t

Molly is selling holiday items for her school fundraiser. Wrapping paper is $3 per roll, chocolate pretzels are $6 per bag, and peppermint candies are $2.50 per bag. When Mr. Netzel placed his order for 5 rolls of wrapping paper, 2 bags of pretzels, and 3 bags of peppermints, he gave Molly $50. Molly needs to find out how much change she owes him. Which answer gives an appropriate checklist to use for keeping this multistep problem organized?

✓ Calculate the amount owed for wrapping paper. ✓ Calculate the amount owed for chocolate pretzels. ✓ Calculate the amount owed for peppermint candies. ✓ Calculate the grand total owed. ✓ Subtract the grand total from $50.