Which expression represents "5 less than 3 times a number n"?
Hint
Find "3 times a number" first. Then "5 less than" that means you take 5 away from it.
Step-by-step solution
"3 times a number n" is 3n.
"5 less than" something means start with that something and subtract 5.
So "5 less than 3n" is 3n - 5.
The expression is 3n - 5.
Check: let n = 4. Three times 4 is 12, and 5 less than 12 is 7. The expression gives 3(4) - 5 = 7 ✓. The choice 5 - 3n would give 5 - 12 = -7, which is wrong.
Tip: "less than" reverses the order of the words. "5 less than x" is x - 5, never 5 - x.
Question 2 of 36
A gym charges a one-time sign-up fee of $35 plus $22 per month. Which expression gives the total cost, in dollars, of a membership for m months?
Hint
Which amount is paid every month? That amount is multiplied by m.
Step-by-step solution
The $22 is charged per month, so for m months it costs 22m.
The $35 sign-up fee is paid once, so it is added just one time.
Tip: the word per marks the rate, which goes with the variable. A one-time fee is the constant.
Question 3 of 36
Drop-down
Select the expression that makes the statement true.
"The quotient of a number x and 4, increased by 9" is written as .
Choose an answer from each menu.
Hint
"The quotient of x and 4" is x divided by 4. "Increased by 9" happens after the division.
Step-by-step solution
"The quotient of a number x and 4" means x \div 4, written \dfrac{x}{4}. The order is the same as the words: x first, then 4.
"Increased by 9" means add 9 to that quotient.
Result: \dfrac{x}{4} + 9.
The expression is \dfrac{x}{4} + 9.
Check: let x = 20. The quotient of 20 and 4 is 5; increased by 9 is 14. The expression gives \dfrac{20}{4} + 9 = 14 ✓. The choice \dfrac{x + 9}{4} gives \dfrac{29}{4} = 7.25 — different.
Tip: "the quotient of A and B" is always A ÷ B. Only use parentheses (or a long fraction bar) when the words group a sum together, as in "the sum of x and 9, divided by 4".
Question 4 of 36
No calculator
Which expression represents "8 more than the product of 4 and a number y"?
Hint
"The product of 4 and y" is one quantity. Write it first, then add 8 to it.
Step-by-step solution
"The product of 4 and a number y" means multiply: 4y.
"8 more than" that product means add 8 to it: 4y + 8.
No parentheses are needed, because the 8 is added after the multiplication.
The expression is 4y + 8.
Check: try y = 5 in words: the product of 4 and 5 is 20, and 8 more than 20 is 28. The expression: 4(5) + 8 = 28 ✓
Tip: "product" signals multiplication and "sum" signals addition. Translate the phrase piece by piece, from the inside out.
Question 5 of 36
A bike-share program charges $2 to unlock a bike plus $0.30 per minute of riding. Which expression gives the cost, in dollars, of a ride that lasts t minutes?
Hint
Which amount is paid once, and which amount is paid for every minute?
Step-by-step solution
The $2 unlock fee is paid once per ride: it is a constant, 2.
The $0.30 is paid for each minute: for t minutes that is 0.30t.
Add the two parts: 2 + 0.30t.
The cost is 2 + 0.30t dollars.
Check: a 10-minute ride should cost \$2 + \$3 = \$5. The expression: 2 + 0.30(10) = 5 ✓
Tip: in cost models, the amount that comes with the word "per" (per minute, per mile, per month) is the number multiplied by the variable.
Question 6 of 36
Fill in
A candle is 30 centimeters tall. It burns down 1.5 centimeters every hour. Its height, in centimeters, after h hours can be written as a + bh.
Enter the values of a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
a is the height before any burning. Because the candle gets shorter, think about the sign of b.
Step-by-step solution
At the start (h = 0), the candle is 30 cm tall. So the constant is a = 30.
Each hour the height goes down by 1.5 cm. After h hours it has lost 1.5h cm.
Height: 30 - 1.5h = 30 + (-1.5)h. So b = -1.5.
The answers are a = 30 and b = -1.5.
Check: after 4 hours the candle has lost 4 \times 1.5 = 6 cm, so it is 24 cm tall. The expression: 30 - 1.5(4) = 24 ✓
Tip: in the form a + bh, a quantity that decreases has a negative b. Ask "does the value go up or down?" before deciding the sign.
Question 7 of 36
The length of a rectangular garden is 3 feet more than twice its width, w, as shown.
Which expression represents the perimeter of the garden, in feet?
Hint
A rectangle has two lengths and two widths. Perimeter P = 2l + 2w is on the formula sheet.
Step-by-step solution
Width = w. Length = 2w + 3 ("3 more than twice the width").
Perimeter of a rectangle: P = 2l + 2w.
Substitute: P = 2(2w + 3) + 2w.
Distribute: 2(2w + 3) = 4w + 6.
Combine: 4w + 6 + 2w = 6w + 6.
The perimeter is 6w + 6 feet.
Check: if w = 4, the length is 2(4) + 3 = 11. Perimeter: 11 + 4 + 11 + 4 = 30. The expression gives 6(4) + 6 = 30 ✓
Tip: perimeter adds all four sides. Adding one length and one width gives only half the perimeter.
Question 8 of 36
Fill in
Sam has $500 in savings. He spends $35 of it each week and adds nothing new. The amount left after w weeks can be written as a + bw.
Enter the values of a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
a is the amount Sam starts with. b is how much the savings change each week — is that change up or down?
Step-by-step solution
The starting amount, which does not depend on w, is $500. So a = 500.
Each week the savings go down by $35. A decrease is a negative rate, so the change per week is -35.
The amount left is 500 + (-35)w = 500 - 35w. So b = -35.
a = 500 and b = -35.
Check: after 2 weeks Sam has spent $70 and has $430 left. The expression gives 500 - 35(2) = 430 ✓
Tip: in the form "fixed amount + rate × quantity", money going out or anything decreasing has a negative rate. Entering b = 35 would describe savings that grow by $35 a week.
Question 9 of 36
The smallest of three consecutive even integers is n. Which expression represents the sum of the three integers?
Hint
Consecutive even integers, such as 4, 6, 8, are 2 apart. Write each one in terms of n.
Step-by-step solution
Consecutive even integers go up by 2: the three integers are n, n + 2 and n + 4.
Add them: n + (n + 2) + (n + 4).
Combine the n-terms: n + n + n = 3n.
Combine the constants: 2 + 4 = 6.
The sum is 3n + 6.
Check: let n = 4. The integers are 4, 6, 8, and their sum is 18. The expression gives 3(4) + 6 = 18 ✓
Tip: consecutive odd integers also go up by 2 (n, n + 2, n + 4). Only plain consecutive integers go up by 1.
Question 10 of 36
Select all
Notebooks cost $3 each and pens cost $1.50 each. Lena buys n notebooks and twice as many pens as notebooks.
Select all expressions that represent the total cost, in dollars.
Select all that apply.
Hint
First write the number of pens in terms of n. Then multiply each quantity by its own price.
Step-by-step solution
Notebooks: n. Pens: twice as many, so 2n.
Cost of notebooks: 3n. Cost of pens: 1.50(2n) = 3n.
Total: 3n + 1.5(2n) = 3n + 3n = 6n.
Now test each choice:
3n + 1.5(2n) ✓ (the unsimplified form)
6n ✓ (fully simplified)
3n + 3n ✓ (pens simplified to 3n)
4.5n ✗ (treats the pens as only n)
7.5n ✗ (doubles the notebooks instead: 3(2n) + 1.5n)
5n + 1.5 ✗ (from 3n + 1.5 + 2n — the price was added, not multiplied)
The correct expressions are 3n + 1.5(2n), 6n and 3n + 3n.
Check: with n = 2, Lena buys 2 notebooks ($6) and 4 pens ($6): $12 total. 6(2) = 12 ✓, but 4.5(2) = 9 and 7.5(2) = 15 ✗
Tip: "twice as many pens as notebooks" doubles the pens. Find which quantity the word twice describes.
Question 11 of 36
A group of 20 people visits a zoo. Adult tickets cost $12 and child tickets cost $7. If a people in the group are adults and everyone else is a child, which expression gives the total cost of the tickets, in dollars?
Hint
If a of the 20 people are adults, how many are children? Write it with a.
Step-by-step solution
Adults: a. Children are the rest of the 20 people: 20 - a.
Cost of adult tickets: 12a.
Cost of child tickets: 7(20 - a).
Total: 12a + 7(20 - a).
The expression is 12a + 7(20 - a). (Simplified, it is 5a + 140.)
Check: if the group has 5 adults and 15 children, the cost is 5(12) + 15(7) = 60 + 105 = 165 dollars. The expression gives 12(5) + 7(15) = 165 ✓
Tip: when a total is split into two groups, one group is x and the other is \text{total} - x, with the total written first.
Question 12 of 36
Drag and drop
Drag the expression that matches each phrase into the box below it. Not every tile is used.
6 fewer than x
6 divided by x
the product of 6 and the sum of x and 2
Tap a tile, then tap a box to place it.
Hint
"Fewer than" reverses the order. "The quotient/divided by" keeps the order. "The sum of" means group it first.
Step-by-step solution
"6 fewer than x": start with x and take away 6 → x - 6. (The tile 6 - x reverses it — that would be "x fewer than 6".)
"6 divided by x": 6 is divided, so 6 goes on top → \dfrac{6}{x}. (The tile \dfrac{x}{6} means "x divided by 6".)
"The product of 6 and the sum of x and 2": find the sum x + 2 first, then multiply by 6 → 6(x + 2). (The tile 6x + 2 multiplies only x.)
Check: using x = 3: 6 fewer than 3 is -3 and 3 - 6 = -3 ✓; 6 divided by 3 is 2 and \tfrac{6}{3} = 2 ✓; 6 times (3 + 2) = 30 and 6(5) = 30 ✓, but 6(3) + 2 = 20 ✗.
Tip: whenever the words say "the sum of" or "the difference of" and then another operation acts on it, you need parentheses.
Question 13 of 36
Maria is 4 years older than twice her son's age. Her son is s years old. Which expression represents the sum of their ages?
Hint
First write Maria's age in terms of s. Then add the son's age to it.
Step-by-step solution
Son's age: s.
"Twice her son's age" is 2s; "4 years older than" that is 2s + 4. This is Maria's age.
Sum of their ages: s + (2s + 4) = 3s + 4.
The sum is 3s + 4.
Check: if the son is 10, Maria is 2(10) + 4 = 24, and the sum is 34. The expression: 3(10) + 4 = 34 ✓
Tip: when a question asks for a total, make sure you have included every person or item. A common slip is to stop after writing Maria's age.
Question 14 of 36
Select all
A school buys n boxes of markers at $9 per box. It buys three times as many boxes of pencils as boxes of markers, at $4 per box.
Select all expressions that represent the total cost, in dollars.
Select all that apply.
Hint
How many boxes of pencils are there in terms of n? Multiply that count by the pencil price.
Step-by-step solution
Markers: n boxes at $9 cost 9n.
Pencils: 3n boxes at $4 cost 4(3n) = 12n.
Total: 9n + 12n = 21n.
Check each choice:
9n + 4(3n) = 21n ✓
9n + 4n + 3 = 13n + 3 ✗
21n ✓
3(9n + 4n) = 39n ✗
9n + 12 ✗
3n(4) + 9n = 21n ✓
The correct expressions are 9n + 4(3n), 21n and 3n(4) + 9n.
Check: with n = 2: 2 marker boxes cost $18 and 6 pencil boxes cost $24, so $42. Then 21(2) = 42 ✓, but 13(2) + 3 = 29, 39(2) = 78 and 9(2) + 12 = 30 ✗
Tip: "three times as many" changes the count of an item (3n), not its price.
Question 15 of 36
A 12-foot board is cut into two pieces. One piece is x feet long. The other piece is then cut into 3 pieces of equal length.
Which expression gives the length, in feet, of each of the 3 equal pieces?
Hint
First write the length of the "other piece". Then divide that whole length by 3.
Step-by-step solution
The board is 12 feet and one piece is x feet, so the other piece is 12 - x feet.
That piece is cut into 3 equal parts, so each part is (12 - x) \div 3.
As a fraction: \dfrac{12 - x}{3}.
Each piece is \dfrac{12 - x}{3} feet long.
Check: if x = 3, the other piece is 9 feet, and each of the 3 parts is 3 feet. The expression: \tfrac{12 - 3}{3} = 3 ✓
Tip: when you divide "the rest", the whole difference must be divided. Use a fraction bar or parentheses: (12 - x) \div 3.
Question 16 of 36
The table shows the total cost of online concert tickets, including a fixed service fee.
Number of tickets, t
1
2
3
4
Total cost ($)
13
21
29
37
Which expression gives the total cost, in dollars, for t tickets?
Hint
How much does the cost go up for each extra ticket? Then work out what the cost would be for 0 tickets.
Step-by-step solution
Find the rate: each extra ticket adds 21 - 13 = 8 dollars (also 29 - 21 = 8 and 37 - 29 = 8). So each ticket costs $8.
Find the fixed fee: go back one step from 1 ticket to 0 tickets: 13 - 8 = 5. The fee is $5.
Expression: 8t + 5.
The expression is 8t + 5.
Check:t = 3: 8(3) + 5 = 29 ✓ and t = 4: 8(4) + 5 = 37 ✓
Tip: the starting value is the output when the input is 0, which is often not in the table. Step back from the first row to find it.
Question 17 of 36
Drop-down
Ms. Ortiz is a math tutor. She uses the expression 45h + 20 to find what she charges, in dollars, for a session that lasts h hours, including her travel.
Select the answers that make the statements true.
In the expression, 20 represents .
In the expression, 45 represents .
Choose an answer from each menu.
Hint
Which number changes the total when the session gets longer, and which number stays the same for every session?
Step-by-step solution
The term 45h grows with the number of hours. So 45 is the amount she charges per hour.
The 20 is added once, no matter how long the session is. It is a fixed amount: the travel charge.
For example, a 2-hour session costs 45(2) + 20 = 110 dollars: $90 for the hours plus $20 travel.
The answers are a one-time travel charge and the amount charged per hour.
Check: a 1-hour session costs $65 and a 2-hour session costs $110. The difference, $45, is the cost of the extra hour ✓
Tip: in an expression like mx + b, the coefficient m is the rate (per unit) and the constant b is the fixed starting amount.
Question 18 of 36
A jacket has a regular price of p dollars. It is on sale for 25% off. Which expression gives the sale price, in dollars?
Hint
If you take 25% off, what percent of the price do you still pay?
Step-by-step solution
The discount is 25% of the price: 0.25p.
Sale price = price minus discount: p - 0.25p.
Combine: p - 0.25p = 1p - 0.25p = 0.75p.
The sale price is 0.75p dollars.
Check: a $60 jacket at 25% off saves $15, so it costs $45. The expression: 0.75(60) = 45 ✓
Tip: "x% off" means you pay (100 - x)%. 25% off → pay 75% → multiply by 0.75. This one-step form is what the GED usually offers as the answer.
Question 19 of 36
Fill in
An internet company charges a one-time installation fee of $49.99 and $34.50 per month for service. The total cost, in dollars, for m months can be written as am + b.
Enter the values of a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
a is multiplied by the number of months. b is paid only once.
Step-by-step solution
The monthly charge is paid every month. For m months it is 34.50m, so a = 34.50.
The installation fee is paid once. It is the constant: b = 49.99.
Total cost: 34.50m + 49.99.
The answers are a = 34.50 and b = 49.99.
Check: for one year (m = 12): 34.50(12) + 49.99 = 414 + 49.99 = 463.99 dollars. Counting directly: 12 months at $34.50 is $414, plus $49.99 installation, so $463.99 ✓
Tip: in am + b, the order of the terms is swapped compared with how the problem was written. Match by meaning (per month vs. one time), not by position.
Question 20 of 36
The sign below is shaped like an isosceles triangle. The base is b inches long, and each of the two equal sides is 2 inches longer than the base.
Which expression represents the perimeter of the sign, in inches?
Hint
The perimeter adds all three sides. Two of them are b + 2.
Step-by-step solution
List the sides: b, b + 2, b + 2.
Add them: b + (b + 2) + (b + 2), or b + 2(b + 2).
Simplify: b + 2b + 4 = 3b + 4.
The perimeter is 3b + 4 inches.
Check: if b = 10, the sides are 10, 12 and 12, and the perimeter is 34. The expression: 3(10) + 4 = 34 ✓
Tip: write each side separately before simplifying. That way you see that the +2 appears twice, giving +4.
Question 21 of 36
A phone plan costs $40 per month and includes 200 text messages. Each text over 200 costs $0.10. Which expression gives the monthly cost, in dollars, for t texts, where t > 200?
Hint
How many of the t texts are extra? Only those are charged $0.10 each.
Step-by-step solution
The first 200 texts are included in the $40.
The number of extra texts is t - 200.
The extra charge is 0.10(t - 200).
Total: 40 + 0.10(t - 200).
Simplify: 40 + 0.10t - 20 = 0.10t + 20.
The monthly cost is 0.10t + 20 dollars.
Check: with 250 texts, 50 are extra and cost 50 \times 0.10 = 5 dollars, so the bill is $45. The expression gives 0.10(250) + 20 = 25 + 20 = 45 ✓
Tip: the simplified form looks strange ("$20 plus 10 cents per text") but it only applies when t > 200. Always check threshold expressions with a number above the limit.
Question 22 of 36
Fill in
A rectangular garden is x feet wide. Its length is 4 feet less than three times its width. Fencing costs $6 per foot.
The cost, in dollars, to fence the whole perimeter of the garden can be written as ax + b. Enter a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write the length first. Find the perimeter with P = 2l + 2w, simplify it, then multiply by the price per foot.
Step-by-step solution
Width = x. Length = "4 less than three times the width" = 3x - 4.
Cost at $6 per foot: 6(8x - 8) = 48x - 48 dollars.
In the form ax + b: a = 48 and b = -48.
a = 48, b = -48.
Check: if x = 10, the garden is 10 ft by 26 ft, so the perimeter is 2(26) + 2(10) = 72 ft and the fence costs 72 \times 6 = 432 dollars. The expression gives 48(10) - 48 = 432 ✓
Tip: build big expressions in layers — sides, then perimeter, then cost — and simplify at each layer. Multiplying by 6 at the end must reach both terms of the perimeter.
Question 23 of 36
A hiker starts at an elevation of 8,400 feet. She rests for the first hour and does not move. After that, she walks downhill and her elevation drops 650 feet per hour.
Which expression gives her elevation, in feet, h hours after she started, where h \ge 1?
Hint
How many of the h hours did she actually spend walking downhill?
Step-by-step solution
She starts at 8,400 feet.
The first hour is a rest, so after h hours she has walked for only h - 1 hours.
In h - 1 hours she drops 650(h - 1) feet.
Elevation: 8{,}400 - 650(h - 1).
The expression is 8{,}400 - 650(h - 1).
Check: at h = 1 she has not moved yet: 8{,}400 - 650(0) = 8{,}400 ✓. At h = 3 she has walked 2 hours: 8{,}400 - 1{,}300 = 7{,}100 feet ✓
Tip: test the expression at the moment something changes (here h = 1). A correct model must give the starting value there.
Question 24 of 36
Fill in
The table shows the amount of water in a tank as it drains at a steady rate.
Time (minutes), m
0
5
10
15
Water (gallons)
240
215
190
165
The amount of water after m minutes can be written as a + bm. Enter a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The time goes up by 5 minutes per column. How much water is lost per one minute?
Step-by-step solution
Starting amount: at m = 0 there are 240 gallons, so a = 240.
Change per column: 215 - 240 = -25 gallons every 5 minutes.
Change per minute: -25 \div 5 = -5 gallons per minute, so b = -5.
Expression: 240 - 5m.
The answers are a = 240 and b = -5.
Check:m = 15: 240 - 5(15) = 240 - 75 = 165 ✓
Tip: when the input goes up by more than 1 per column, divide the change in output by the change in input to get the rate. Using -25 as the rate is a very common trap.
Question 25 of 36
The expression 1{,}200 - 85w gives the amount, in dollars, that Kofi still owes on a loan after he makes w weekly payments.
Which statement is true?
Hint
What happens to the amount owed each time w goes up by 1? What is the amount when w = 0?
Step-by-step solution
When w = 0 (no payments yet), he owes 1{,}200 - 0 = 1{,}200 dollars. So he borrowed $1,200.
Each extra payment lowers the amount by 85 dollars. So each weekly payment is $85.
Check the other ideas: he does not pay $1,200 a week, and the loan is paid off when 1{,}200 - 85w = 0, which is about 14.1 weeks, not 85 weeks.
The true statement is Kofi pays $85 each week.
Check: after 2 payments he owes 1{,}200 - 170 = 1{,}030 dollars, which is $1,200 minus two payments of $85 ✓
Tip: to interpret an expression, plug in 0 for the variable (that gives the starting value) and then 1 (that shows the change per step).
Question 26 of 36
Fill in
At a restaurant, the food bill is b dollars. Sales tax of 8% of the food bill is added, and Jada also leaves a tip of 15% of the food bill (before tax).
The total amount Jada pays, in dollars, can be written as kb. What is the value of k?
k =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write the total as b + (\text{tax}) + (\text{tip}), where both the tax and the tip are a percent of b. Then combine like terms.
Step-by-step solution
Tax: 8% of b is 0.08b.
Tip: 15% of b is 0.15b.
Total: b + 0.08b + 0.15b.
Combine like terms: 1 + 0.08 + 0.15 = 1.23, so the total is 1.23b.
The value is k = 1.23.
Check: a $40 food bill: tax $3.20, tip $6.00, total $49.20. The expression: 1.23(40) = 49.20 ✓
Tip: don't forget the "1" for the original bill. A total that includes the original amount always has a coefficient bigger than 1.
Question 27 of 36
Drag and drop
Three friends share the driving on a road trip. Ana drives d miles. Ben drives 40 miles more than Ana. Cy drives half as far as Ben.
Drag an expression into each box. Not every tile is used.
Miles Ben drives
Miles Cy drives
Total miles driven
Tap a tile, then tap a box to place it.
Hint
Write Ben's distance first. Cy's distance is half of all of Ben's distance.
Step-by-step solution
Ben: 40 more than Ana: d + 40.
Cy: half of Ben's distance: \tfrac{1}{2}(d + 40) = 0.5d + 20, so 0.5d + 20.
Total: d + (d + 40) + (0.5d + 20) = 2.5d + 60, so 2.5d + 60.
Unused tiles: 40d (reads "40 more than" as "40 times") and 0.5d + 40 (halves d but not the 40).
Check: if Ana drives 100 miles, Ben drives 140 and Cy drives 70, for 310 in all. The tiles: 100 + 40 = 140 ✓, 0.5(100) + 20 = 70 ✓, 2.5(100) + 60 = 310 ✓
Tip: "half as far as Ben" means halve Ben's whole distance. Put parentheses around d + 40 before you take half.
Question 28 of 36
At a fundraiser, adult tickets cost $15 and student tickets cost $6. The number of student tickets sold was 25 fewer than three times the number of adult tickets, a.
Which expression gives the total money collected from ticket sales, in dollars?
Hint
First write the number of student tickets in terms of a. Then multiply each count by its price.
Step-by-step solution
Adult tickets: a, bringing in 15a dollars.
Student tickets: "three times the adult tickets" is 3a; "25 fewer than" that is 3a - 25.
Student money: 6(3a - 25) = 18a - 150.
Total: 15a + 18a - 150 = 33a - 150.
The total is 33a - 150 dollars.
Check: if a = 20, there are 3(20) - 25 = 35 student tickets. Money: 15(20) + 6(35) = 300 + 210 = 510. The expression: 33(20) - 150 = 510 ✓
Tip: "25 fewer than three times a" is 3a - 25, not 25 - 3a. "Fewer than" and "less than" reverse the order you hear.
Question 29 of 36
Fill in
A test has 50 questions, and every question is answered. A student gets c questions correct and the rest wrong. Each correct answer earns 2 points, and each wrong answer loses 0.5 point.
The student's score can be written as ac + b. Enter a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
How many questions are wrong, in terms of c? Write the score, then distribute and combine.
Step-by-step solution
Correct answers: c, worth 2c points.
Wrong answers: the rest, 50 - c, each worth -0.5: -0.5(50 - c).
Score: 2c - 0.5(50 - c) = 2c - 25 + 0.5c.
Combine: 2c + 0.5c = 2.5c. Score = 2.5c - 25.
The answers are a = 2.5 and b = -25.
Check: if the student gets 40 correct and 10 wrong: 80 - 5 = 75 points. The expression: 2.5(40) - 25 = 75 ✓
Tip: "the rest" of a fixed total T is always T - x. Put it in parentheses so the number in front multiplies both parts.
Question 30 of 36
Mia earns $18 per hour for the first 40 hours she works in a week. For each hour over 40, she earns 1.5 times her regular hourly rate.
Which expression gives Mia's pay, in dollars, for a week in which she works h hours, where h > 40?
Hint
Find the overtime rate first. Then split the week into the first 40 hours and the h - 40 overtime hours.
Step-by-step solution
Overtime rate: 1.5 \times 18 = 27 dollars per hour.
Pay for the first 40 hours: 18 \times 40 = 720 dollars.
Check: for 5,000 gallons: 2,000 extra gallons cost 2{,}000 \times 0.004 = 8, so the bill is 18 + 8 = 26. The expression: 0.004(5{,}000) + 6 = 20 + 6 = 26 ✓
Tip: on the GED, the answer to a threshold problem is often shown in simplified form. If no choice matches your unsimplified expression, distribute and combine.
Question 32 of 36
Select all
A food truck sells sandwiches. Its daily profit, in dollars, when it sells s sandwiches is
7.5s - 2.25s - 180
where 7.5s is the money from sales, 2.25s is the cost of ingredients, and 180 is a fixed daily cost.
Select all statements that are true.
Select all that apply.
Hint
Think about what each term means. Then simplify: 7.5s - 2.25s = 5.25s. Try s = 0 as a test.
Step-by-step solution
7.5s is the money from sales, so each sandwich sells for $7.50. Statement "sells for $2.25" is false - $2.25 is the ingredient cost.
The 180 is subtracted no matter how many sandwiches are sold: a fixed daily cost of $180. True.
Simplify: 7.5s - 2.25s = 5.25s. Each sandwich adds $5.25 of profit before the fixed cost. True.
Break-even means profit = 0: 5.25s = 180, so s \approx 34.3, meaning about 35 sandwiches - not 180 sandwiches. False.
At s = 0: profit = -180, a loss of $180, not $0. False.
The true statements are "The truck has $180 in fixed costs each day", "The profit from each sandwich before fixed costs is $5.25" and "Each sandwich sells for $7.50".
Tip: constants in a profit expression are usually fixed costs (subtracted) or fixed income (added). Coefficients are amounts per item.
Question 33 of 36
A store buys a lamp for c dollars. It marks the price up by 40%, and later puts the lamp on sale for 20% off the marked-up price.
Which expression gives the sale price, in dollars?
Hint
Do one change at a time. A 40% markup multiplies by 1.40. Then 20% off multiplies that price by 0.80.
Step-by-step solution
Marked-up price: c + 0.40c = 1.40c.
Sale price: 20% off means paying 80% of the marked-up price: 0.80(1.40c).
Multiply: 0.80 \times 1.40 = 1.12. Sale price = 1.12c.
The sale price is 1.12c dollars.
Check: if the lamp costs the store $50: markup to $70, then 20% off $70 is $14, so the sale price is $56. The expression: 1.12(50) = 56 ✓
Tip: percent changes in a row multiply; they do not add. A 40% increase followed by a 20% decrease is not a 20% increase, because the 20% is taken from a bigger number.
Question 34 of 36
A banquet hall charges a fixed room fee plus a cost per guest. The table shows the total charge for three events.
Guests, g
10
25
40
Total charge ($)
365
680
995
Which expression gives the total charge, in dollars, for g guests?
Hint
The number of guests goes up by 15 between columns. Divide the change in charge by 15 to get the cost per guest.
Step-by-step solution
Change in guests: 25 - 10 = 15. Change in charge: 680 - 365 = 315.
Cost per guest: 315 \div 15 = 21 dollars. (Also 995 - 680 = 315 for the next 15 guests ✓)
Room fee: for 10 guests the charge is 365, and the guests cost 21 \times 10 = 210. Fee: 365 - 210 = 155.
Tip: the first column of a table is not always g = 0 or g = 1. Find the rate first, then subtract (rate × first input) from the first output to get the fixed fee.
Question 35 of 36
Fill in
A carpet cleaner charges $40 per visit plus $0.35 per square foot of carpet cleaned. A rectangular room is 12 feet wide and x feet long, and its whole floor is carpeted.
The cost, in dollars, to clean the carpet in this room can be written as ax + b. Enter a and b.
a =
b =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
First write the area of the room in terms of x. The price per square foot multiplies the area.
Step-by-step solution
Area of the room: 12 \cdot x = 12x square feet.
Cleaning charge: 0.35 \times 12x = 4.2x dollars.
Add the visit fee: 4.2x + 40.
The answers are a = 4.2 and b = 40.
Check: a room 12 ft by 15 ft has 180 square feet. Cost: 40 + 0.35(180) = 40 + 63 = 103. The expression: 4.2(15) + 40 = 63 + 40 = 103 ✓
Tip: when a rate is "per square foot", you need an area first. Watch for units: a rate per square foot times a length alone gives the wrong unit.
Question 36 of 36
Which situation could be represented by the expression 3(x + 4)?
Hint
In 3(x + 4), you add 4 to xfirst and then multiply the result by 3. Find the situation that works in that order.
Step-by-step solution
3(x + 4) means: start with x, add 4, then multiply the result by 3.
"3 movie tickets that each cost $4 more than x dollars": each ticket costs x + 4, and 3 tickets cost 3(x + 4) ✓
"3 tickets at x dollars each plus a $4 fee" is 3x + 4: the 4 is added once, after multiplying.
"Three people split a bill of x + 4 dollars" is \tfrac{x + 4}{3}: splitting means dividing.
"4 tickets at x + 3 dollars each" is 4(x + 3): the numbers are swapped.
The answer is the cost of 3 movie tickets that each cost $4 more than a ticket costing x dollars.
Check: with x = 10: each ticket costs $14 and 3 tickets cost $42. 3(10 + 4) = 42 ✓, while 3(10) + 4 = 34 ✗
Tip: parentheses in an expression mean "this whole amount happens to each one". Look for a situation where the +4 belongs to every item.
