Algebra Word Problems & Function Interpretation - MCQs Practice 2
Answers will be provided at the end
1. The total number of pages in a book is T. Eloise will read P pages each day for exactly D days until she completes the book at the end of the final day. Which of the following expresses the relationship between these values?
Since she will read P pages each day for D days, multiply P and D together to get the total number of pages read: T = PD.
2. Mary rented a boat that cost a $100 flat fee plus $3 for each gallon of gasoline used. If Mary paid a total of $130 to rent the boat, how many gallons of gasoline did she use?
Using x as the number of gallons of gasoline used, set up an equation modeling the situation like this:
130 = 100 + 3x
This means that the $100 flat fee plus $3 times each gallon of gas used will equal a total amount of $130. Then, solve for x:
130 = 100 + 3x → 30 = 3x → x = 10.
So, 10 gallons of gas were used.
3. An average pace for a backpacker on a certain trail is to hike 10 miles per day. If Joseph is taking a weeklong backpacking trip in which he wants his average pace over the entire week to exceed this average pace each day, which inequality represents the number of miles, m, he will travel on his trip?
If Joseph exceeds the average pace of 10 miles per day over the 7-day period, he will travel in excess of 10 × 7 = 70 miles all together. Express this as an inequality by making the number of miles, m, be greater than 70: m > 70.
4. A bread recipe calls for 6 eggs for every 10 cups of flour. If Martin wants to use a total of 15 eggs, how many cups of flour will he need to make this larger amount of bread using this recipe?
Set up a proportion to solve this. There are 6 eggs for every 10 cups of flour, and this ratio needs to equal the ratio when there are 15 eggs:
Flour/Eggs: 10/6 = x/15 → Cross multiply to solve for x → (10/6) × 15 = x → x = 25.
5. The number of windmills at a windfarm starts at 200, with 6 new windmills added each year and 2 mills removed each year due to wear and tear. What function represents the number of windmills, w, on the windfarm t years after the windfarm opens?
Since the number of windmills begins at 200, this number can be a constant. Each year, a net of 4 windmills are added to the windfarm, since 6 are added and 2 taken away. Therefore, the function that portrays the situation is: w(t) = 200 + 4t.
6. A catalog weighs enough that it requires 4 stamps for the post office to mail it. If a business is sending 500 catalogs, and each book of stamps has 20 stamps in it, how many books of stamps would the business need to purchase to mail all the catalogs?
The business will need to purchase a total of 4 stamps for each of 500 catalogs, so to get the total number of stamps needed, multiply 4 and 500 together: 4 × 500 = 2,000.
Then, calculate the number of stamp books needed by taking the total of 2,000 stamps and dividing it by the number of stamps in each stamp book, 20: 2,000 / 20 = 100.
7. Each student on a school trip packs 2 suitcases, 1 backpack, and 1 personal object. If there are 65 students on the trip, how many total items would the students pack?
Each student packs a total of 2 + 1 + 1 = 4 personal items. Therefore, simply multiply 4 by 65 to find the total items the students would pack: 65 × 4 = 260.
8. The cost in dollars, C, of installing a sewer pipe that is L feet long is modeled by the function below:
C(L) = 300 + 20L
Given that function, what is the most logical interpretation of the following equation?
C(20) = 700
The 20 goes in the place of the variable L, which represents the length of the pipe. Therefore, the length of the pipe in consideration is 20 feet. C(L) represents the cost to install a pipe of length L, so 700 is the cost of installation for a pipe of 20 feet.
9. A plumber's compensation is $50 per hour plus a $5,000 profit-share at the end of the year. Which inequality represents the number of hours, n, she will need to work in order to make at least $50,000 total compensation for the year?
The plumber's compensation is the sum of $5,000 plus $50 per each of n hours worked. This total compensation needs to be at least $50,000, meaning the sum of the hourly wages and the profit sharing needs to be greater than or equal to 50,000. 50,000 ≤ 50n + 5,000 represents this situation.
10. The height in feet of an airplane, A, and a balloon, B, are modeled by these two equations, in which t is the number of minutes after 12:00 P.M.
A = 10,000 - 180t
B = 20t
At what time that same day would the airplane and the balloon be at the same height?
Set them equal to each other and solve for t:
10,000 - 180t = 20t → 10,000 = 200t → t = 50.
So, take 50 minutes past 12:00 to get 12:50 P.M.
1. If a cat eats x cups of food per day, how many cups of food will the cat eat in y days?
Take the x cups of food and multiply it by the y days the cat eats it: x × y = xy.
2. The relationship between distance, D, rate, R, and time, T, is modeled by the equation D = RT. If the rate increases but the distance remains the same, what must happen to the time?
Isolate time: T = D / R. If the rate increases while distance stays the same, the fraction decreases because the denominator grows. Therefore, time decreases.
3. The average price of a gallon of milk is modeled by the function C, in which y is the number of years after the year 2000.
C(y) = 2.00 + 0.08y
What does the number 0.08 in the function represent?
In the function, 2.00 is the base price in the year 2000. Each year after, the price increases by 0.08. Thus, 0.08 represents the average yearly increase in price per gallon.
4. Out of the 50 states in the United States, all but 1, Nebraska, have bicameral legislatures—i.e., there are two legislative chambers. Nebraska has a unicameral legislature—i.e., there is just one legislative chamber. What fraction of the states in the United States are bicameral?
Since 1 state is unicameral, 50 - 1 = 49 states are bicameral. The fraction is 49 out of 50 states, or 49/50.
5. Tabitha works as a cell phone salesperson. Her pay is $10 per hour plus $20 commission for each new cell phone she sells. If H represents the number of hours she works in a given week and C represents the number of cell phones she sells that week, which equation represents her total salary (expressed in dollars before any deductions for taxes, etc.) for the week?
Tabitha makes $10 for each of the H hours and $20 for each of the C cell phones. Total structure: Salary = 10H + 20C.
6. A movie theater concession counter sells a large popcorn for $5 and a small popcorn for $3. If the total number of both types of popcorn sold in a given day is 150, and the total revenue from the popcorn sales is $570, how many small popcorns were sold? (Ignore sales tax in your calculations.)
Let L = large and S = small. L + S = 150 → L = 150 - S.
Revenue equation: 5L + 3S = 570.
Substitute: 5(150 - S) + 3S = 570 → 750 - 2S = 570 → -2S = -180 → S = 90.
7. The cost in dollars to operate a lemonade stand, C, is given by the function c(x) = .25x + 20 in which x represents the number of cups of lemonade sold. What does the number 20 represent in this situation?
If 0 cups are sold (x = 0), the cost is still $20. This flat overhead value represents the initial, fixed baseline setup cost.
8. The linear function f(x) = kx + b has constants k and b. What is the y-intercept of the function based on the values of x and f(x) in the table below?
| x | f(x) |
|---|---|
| 16 | -28 |
| 21 | -38 |
Slope k = (-38 - (-28)) / (21 - 16) = -10 / 5 = -2.
Using f(x) = -2x + b with point (16, -28):
-28 = -2(16) + b → -28 = -32 + b → b = 4. The y-intercept is 4.
9. John has $20 to spend on his school lunch each week. A meal costs $3 and a milk costs $0.50. If he purchases one meal each day on Monday through Friday, what is the greatest number of milks he could purchase that week given his budget (note that sales tax is not applied to his purchases)?
5 meals cost 5 × 3 = 15 dollars. Remaining budget: 20 - 15 = 5 dollars. Milks calculation: 5 / 0.50 = 10 milks.
10. In store A, 3 out of every 4 items are sold at a discount. In store B, 1 out of every 3 items is sold at a discount. If each store has 240 total items it is selling (both full price and at a discount), how many more discounted items will there be in store A than in store B?
Store A items: (3/4) × 240 = 180. Store B items: (1/3) × 240 = 80.
Difference: 180 - 80 = 100.
1. If 2x + k = 2(x + 3), what is the value of the constant k?
Expand right side: 2x + k = 2x + 6. Subtracting 2x from both sides yields k = 6.
2. The price of a painting, p, is modeled by the equation p = 800 + 20y, in which y is the number of years after the painting was created. What is the most logical interpretation of the number 20 in the equation?
When y increases by 1 year, the valuation increments by 20. Therefore, 20 is the recurring yearly increase in price.
3. Rental of a video game costs a flat fee of $20 plus $3 for each hour the video game is used. Which equation provides the total cost, C, of using the game for H hours?
Flat baseline is 20, variable rate is 3 per hour. Combined: C(H) = 3H + 20.
4. The table above provides pairs of values for x and y. Which equation below correctly shows the linear relationship between these values?
| x | y |
|---|---|
| 6 | 0 |
| 4 | -2/3 |
| 2 | -4/3 |
| 0 | -2 |
| -2 | -8/3 |
From the table context, when x = 0, y = -2 (y-intercept). Looking at choices, only option (A) exhibits a y-intercept of -2.
5. A baker sells cupcakes for $3 each and cookies for $2 each. If he sells a total of 200 cookies and cupcakes and the total amount of revenue from selling the cookies and cupcakes is $460, which set of equations could be used to solve for x cupcakes sold and y cupcakes sold?
Total count equation: x + y = 200. Revenue generation framework based on unit prices: 3x + 2y = 460.
6. A discount bookstore sells books for $6 each and magazines for $3 each. In a given week, the store sells a combined total of 750 books/magazines and makes a total of $3,600 from the book and magazine sales. In that week, how many magazines were sold?
Equations: B + M = 750 → B = 750 - M and 6B + 3M = 3600.
Substitution step: 6(750 - M) + 3M = 3600 → 4500 - 3M = 3600 → -3M = -900 → M = 300.
7. A house is initially purchased for $150,000 and increases in value by $3,500 each year. Which of these expressions gives the price, P, of the house t years after purchase?
Initial y-intercept value is 150,000, and annual scaling progression tracking rate is 3,500. Arranged: P = 150,000 + 3,500t.
8. In the following equation, 3x + 5 = 3x + k what must be true about the constant k equal in order for there to be no real solutions for x?
Subtracting 3x yields 5 = k. If k equals 5, there are infinite solutions. For the statement to form a contradiction with no solutions, k ≠ 5.
9. In an art class, a bin contains C crayons and M markers. If there are twice as many markers as crayons, and the total number of markers and crayons in the bin is 90, which of these sets of equations could be used to solve for C and M?
Sum is 90: C + M = 90. "Twice as many markers as crayons" translates directly into M = 2C.
10. Chris took a bike ride through the hills near his home. On his ride, he either went uphill or downhill. His total ride was 30 miles and took 1.6 hours. If half of the distance of his journey was uphill and the other half of the distance was downhill, and his downhill speed was an average of 25 mph, what was his average uphill speed?
Each section distance is 15 miles. Downhill travel evaluation time via t = D/R yields: 15 / 25 = 0.6 hours. Remaining uphill window calculation: 1.6 - 0.6 = 1.0 hour. Speed required to finish 15 miles within a single hour is 15 / 1 = 15 mph.