Free math worksheets

In the United States, temperature is commonly measured in degrees Fahrenheit. To convert a temperature from degrees Celsius to degrees Fahrenheit, multiply the Celsius value by \(1.8\) and add \(32\). a) Let \(c\) be the temperature in degrees Celsius and \(F\) be the temperature in degrees Fahrenheit. Write an equation that models the relationship. b) Use the equation to convert \(20\,^{\circ}\text{C}\), \(35\,^{\circ}\text{C}\), and \(-10\,^{\circ}\text{C}\) to degrees Fahrenheit.

Write and solve an equation for each number riddle. a) What number must be increased by \(2.5\) to get \(1\frac{1}{4}\)? b) What number gives \(-1.5\) when \(\frac{2}{3}\) is subtracted from it? c) Three times what number is \(12.6\)?

Three piles of counters are on a table: a left pile \(L\), a middle pile \(M\), and a right pile \(R\). The left pile has at least \(6\) counters. The middle pile starts with \(z\) counters. Follow these directions: 1. Move \(6\) counters from the left pile to the middle pile. 2. Move \(4\) counters from the middle pile to the right pile. a) Write a simplified expression for the number of counters in the middle pile after both moves. b) The middle pile then has \(15\) counters. Write and solve an equation for its starting amount.

A gym charges a one-time enrollment fee of \(\$s\) and a monthly fee of \(\$m\). Let \(G\) be the total membership cost, in dollars, for \(x\) months. a) Write an equation that models \(G\). b) Find the total cost for one year when \(s=35.00\) and \(m=19.90\).

Write and solve an equation for each statement. 1) A number \(x\) is increased by \(8.4\), and the result is \(15\). 2) A number \(y\) is subtracted from \(42\), and the result is \(13\frac{1}{2}\). 3) Three times a number \(a\) is \(25.5\).

For values of \(x\) and \(y\) that make all lengths positive and form a valid triangle, a student claims that these figures have equal perimeters: A square with side length \(s=1.5x+2y\) A triangle with side lengths \(2x+3y\), \(3x+4y\), and \(x+y\) Check the claim by writing and simplifying an expression for each perimeter.

For values of \(x\) and \(y\) that make the lengths positive and form a valid triangle, the side lengths are \(5x+2y\), \(3x-4y\), and \(2x+3y\). Write and simplify an expression for the perimeter \(P\).

For values of \(x\) and \(y\) that make the lengths positive and form a valid quadrilateral, its side lengths are \(s_1=3x+2\) \(s_2=2y-1\) \(s_3=x+y+4\) \(s_4=2x-y+3\). Write and simplify an expression for the perimeter \(P\).

Anna, Beth, and Clara collect a total of \(120\,\text{lb}\) of paper for a school recycling drive. Anna collects \(10\,\text{lb}\) more than Beth and \(10\,\text{lb}\) less than Clara. Let \(x\) be Anna’s amount. Write an equation for the three amounts, simplify it, solve it, and report how many pounds each person collects.

Lucas, Mia, and Noah have \(200\) trading cards altogether. Lucas has \(20\) more cards than Mia, and Noah has twice as many cards as Mia. Let \(x\) be Mia’s number of cards. Write one equation for the total, solve it, and report each person’s number of cards.

A streaming service offers two monthly plans. Basic: a monthly fee of \(\$G\) plus \(\$k\) for each rented movie. Premium: a flat monthly fee of \(\$P\) with no additional charge per movie. Let \(K_P\) be the Premium plan's monthly cost. a) Write an expression for the Basic plan's monthly cost \(K_B\) when a customer rents \(n\) movies. b) Write an expression for the difference \(K_P-K_B\). c) What does it mean when the expression from part b) equals \(0\)?

A class rents a bus. The total cost \(K\) consists of a fixed fee \(G\) plus \(p\) dollars for each of \(n\) participants: \( K=G+np. \) a) Rearrange the formula to solve for \(n\). b) Find \(n\) when \(K=468\), \(G=120\), and \(p=14.50\). c) Rearrange the original formula to solve for \(p\).

A student has already earned \(26\) service points. Each community event adds \(9\) points. The student needs at least \(80\) points to qualify for an award. Write and solve an inequality for the number \(e\) of additional events needed. Interpret the solution.

A storage shelf can hold no more than \(38\,\text{lb}\). A fixed container weighs \(14\,\text{lb}\), and each identical supply box weighs \(4\,\text{lb}\). Write and solve an inequality for the number \(b\) of supply boxes that can be placed on the shelf. What is the greatest possible number of boxes?

A bookshelf is \(\frac13\) full. After \(12\) more books are added, it is \(\frac12\) full. Let \(x\) be the number of books the shelf holds when full. a) Write an equation with \(x\) on both sides that models the situation. b) Solve your equation algebraically. Your work must show equivalent equations that remove the fractional coefficients and then isolate \(x\). c) Interpret the solution in the context.

A cookbook gives this rule for roasting meat: “Allow \(45\) minutes per pound, plus \(20\) additional minutes for preheating and resting.” a) Write an equation for the total time \(T\), in minutes, for a roast weighing \(m\) pounds. b) Find the total time for a \(2.5\)-pound roast. c) A roast takes \(155\) minutes in all. Use your equation to find its weight.

Posters will be hung along a \(44\,\text{ft}\) wall. Each poster is \(4\,\text{ft}\) wide. All gaps—including the gaps from the end posters to the ends of the wall—must have the same width. a) What is the width of each gap when \(4\) posters are hung? b) Let \(x\) be the number of posters when every gap is exactly \(4\,\text{ft}\) wide. Write an equation that models the wall length, solve it, and state how many posters fit.

Square posters with side length \(2.5\,\text{ft}\) will be hung in one row along a \(43\,\text{ft}\) school hallway wall. Equal gaps of width \(x\) will be left between adjacent posters and at both ends of the row. a) Explain why the total-length model for \(n\) posters is \(L=2.5n+x(n+1)\). b) Exactly \(12\) posters will be used. Find \(x\) when the entire wall is filled. c) How would the model change if the posters touched each other but the row had a fixed \(4\,\text{ft}\) margin at each end?

A water tank starts with \(1000\) gallons. Because of a leak, it loses water at a constant rate of \(15\) gallons per hour. a) Write an expression for the amount of water remaining after \(t\) hours. b) Use the expression to find how much water remains after \(12\) hours. c) Write and solve an equation for the time when the tank is empty. Round to the nearest tenth of an hour.

A bank statement shows a \(\$450.00\) rent payment and a \(\$1250.60\) paycheck deposit. After both transactions, the account balance is \(\$840.25\). Write and solve an equation to find the account balance immediately before the two transactions.

A hot-air balloon rises \(245\,\text{m}\) and then descends \(312\,\text{m}\). Its final altitude is \(128\,\text{m}\). Write and solve an equation to find the balloon’s altitude before these two changes.

A rectangular garden bed has width \(x\) meters. Its length is \(3\,\text{m}\) more than twice its width. a) Write an expression for the perimeter \(P\). b) Simplify the expression. c) Find the total fence length needed when the width is \(4.5\,\text{m}\).

A row of adjacent squares is built with matchsticks. The first square uses \(4\) matchsticks. Each additional square shares one side with the square before it, so it requires only \(3\) more matchsticks. a) Write an equation for the number of matchsticks \(S\) needed for \(n\) squares. b) How many matchsticks are needed for \(15\) squares? c) A row uses \(76\) matchsticks. How many squares are in the row?

A wireframe model of a rectangular prism is made from \(20\,\text{ft}\) of wire. One base edge is twice the length of the other base edge, and the height equals the longer base edge. Let \(x\) be the shorter base edge. Write a one-variable equation for the total length of all \(12\) edges, solve it, and report the three edge lengths.

A wireframe square pyramid is made from a \(160\,\text{in.}\) aluminum rod. Its four base edges have equal length. Each of the four edges from the base to the apex is exactly \(5\,\text{in.}\) longer than a base edge. Let \(a\) be a base-edge length. Write an equation for the total rod length, solve it, and report both edge lengths.

A rectangular poster is \(10\,\text{in.}\) wide. If the width is increased by \(5\,\text{in.}\) and the height is decreased by \(2\,\text{in.}\), the area increases by \(10\,\text{in.}^2\). Let \(h\) be the original height. Write an equation with \(h\) on both sides that compares the new area with the original area, solve it algebraically, and report the original height.

Paul is selling \(40\) used video games at a flea market. His total cost for the booth and the games was \(\$120.00\). He has already sold \(15\) games for \(\$2.50\) each. At what price must he sell each of the remaining games to recover exactly his total cost? Write and solve an equation.

A student newspaper club prints \(200\) copies of a new issue. Printing costs \(\$180.00\), and the club wants to earn an additional \(\$60.00\) for the class fund. The first \(120\) copies were sold for \(\$1.00\) each. At what price must the remaining copies be sold? Write and solve an equation.

A raffle has \(500\) tickets. The original plan was to sell every ticket for \(\$1.50\). During a promotion, \(300\) tickets were sold for an average of \(\$1.10\) each. What price must be charged for each of the remaining \(200\) tickets to reach exactly the originally planned total revenue? Write and solve an equation.

Fresh mushrooms are \(92\%\) water. A package initially contains \(500\,\text{g}\) of mushrooms. The mushrooms are dried until they are only \(20\%\) water. What is their new total mass?

A pot contains \(2\,\text{L}\) of a fruit drink that is \(25\%\) pure juice and \(75\%\) water. The drink is heated, and only water evaporates. After heating, the mixture is \(40\%\) juice. How much water evaporated?

A rectangle has a perimeter of at most \(48\,\text{in.}\). One side is exactly three times as long as the other side. Let \(x\) be the shorter side. Write and solve an inequality for the perimeter, then give the greatest possible length of each side.

A rectangular flower bed can use at most \(40\,\text{ft}\) of edging. Its length will be exactly \(4\,\text{ft}\) greater than its width. Let \(w\) be the width. Write and solve an inequality for the edging constraint, then state the greatest possible width.

A small birdbath starts with \(15\,\text{L}\) of water. During a rain shower, \(0.5\,\text{L}\) flows in each minute. At the same time, a leak drains a constant amount \(y\), in liters per minute. After 20 minutes, the birdbath contains exactly \(21\,\text{L}\). A student proposes the equation \(15 + 0.5 \cdot 20 - y \cdot 20 = 21\). a) Does the equation correctly model the situation? Briefly explain. b) Find the water loss \(y\) in liters per minute.

Lucas spends \(\frac{1}{4}\) of his monthly allowance on movie tickets and \(\frac{2}{5}\) on a new video game. He puts the remaining \(\$14\) into savings. How much allowance did Lucas receive in all?

A school library receives a shipment of new books. One-half are novels, one-fifth are nonfiction books, and one-eighth are graphic novels. The remaining \(63\) books are children’s books. Write and solve an equation to find the total number of new books \(x\). Briefly explain what \(1 - \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{8}\right)\) represents in this situation.

A mental math performer at a carnival says, “Choose a secret number, double it, add \(10\), and divide the result by \(2\). Tell me your final result.” a) Let the starting number be \(x\). Write and simplify an expression for the final result. b) One participant says the final result is \(42\). What was the starting number? c) Describe the quick mental rule the performer can use to recover the starting number.

There are three piles of counters. The left and right piles each start with exactly \(12\) counters. The middle pile starts with \(z\) counters. Follow these steps: 1. Move \(5\) counters from the left pile to the middle pile. 2. Move \(3\) counters from the right pile to the middle pile. 3. Count the counters remaining in the left pile. Remove that same number of counters from the middle pile and set them aside. a) How many counters remain in the left pile after step \(1\)? b) Write and simplify an expression for the number of counters in the middle pile after step \(3\). c) How many counters are in the middle pile at the end if \(z=10\)?

A large fruit crate contains only apples and pears. Exactly \(65\%\) of the fruit are apples, and there are \(42\) more apples than pears. How many pieces of fruit are in the crate altogether, and how many are pears?

Write an equation and solve the number riddle: “Twice a number \(x\), increased by the difference between \(15.4\) and \(8.4\), equals \(25\).” What is \(x\)?

A bag contains red, yellow, and white gummy bears. There are exactly \(12\) red gummy bears, and the probability of selecting a red gummy bear is \(0.25\). a) Find the total number of gummy bears. b) How many yellow and white gummy bears are there altogether? c) There are twice as many yellow gummy bears as white gummy bears. Find the number of yellow gummy bears.

Two bags contain colored counters. Bag A contains \(5\) blue and \(15\) red counters. Bag B contains only green and yellow counters. The probability of drawing a red counter from Bag A equals the probability of drawing a green counter from Bag B. a) Bag B contains \(12\) green counters. Find the number of yellow counters in Bag B. b) All counters from both bags are combined. Find the probability of drawing a red counter from the combined container.

A school raffle has grand-prize tickets, small-prize tickets, and losing tickets. The probability of a grand prize is \(\frac{1}{50}\), and the probability of a small prize is \(\frac{1}{5}\). The raffle box contains exactly \(390\) losing tickets. a) How many tickets were prepared altogether? b) How many grand-prize tickets are there?

A container holds \(40\) red, blue, and green counters. It contains \(15\) red counters and \(10\) blue counters. a) Find the probability of drawing a green counter. b) Find the probability of not drawing a blue counter. c) Some red counters are removed. Afterward, the probability of drawing a blue counter is exactly \(\frac{1}{3}\). How many red counters were removed?

Max and Julia start \(18\) miles apart and walk toward each other. Max walks at \(4\,\text{mph}\). Julia walks \(50\%\) faster than Max. They start at the same time. Use a linear equation to determine who travels farther before they meet and by how many miles.

A composite banner is made from a parallelogram and a triangle that share an edge but do not overlap. The parallelogram has base \(8\,\text{in.}\) and height \(4.5\,\text{in.}\). The triangle has base \(12\,\text{in.}\). The entire composite banner has area \(66\,\text{in.}^2\). Find the triangle's height.

A rectangular garden is \(12\,\text{ft}\) long and has an area of \(120\,\text{ft}^2\). A triangular flower bed is attached along one of the garden's shorter sides, so the triangle's base equals the garden's width. The combined area is \(150\,\text{ft}^2\). Find the height of the triangular flower bed.

Two classes sell raffle tickets for a fundraiser. Class A has \(120\) tickets, including \(30\) winning tickets. Class B has \(150\) tickets, including \(40\) winning tickets. a) Which class has the greater probability of drawing a winning ticket? Compare the probabilities. b) How many losing tickets must be added to Class B so that both classes have exactly the same winning probability?

Complete the account table. In each row, an amount is withdrawn from the previous balance to produce the new balance. <table> <tr><th>Previous balance</th><th>Withdrawal</th><th>New balance</th></tr> <tr><td>\(\$12\)</td><td>\(\$20\)</td><td>(1)</td></tr> <tr><td>\(-\$8\)</td><td>\(\$15\)</td><td>(2)</td></tr> <tr><td>(3)</td><td>\(\$25\)</td><td>\(-\$10\)</td></tr> </table>

Ms. Weber’s bank account balance is \(-\$120\). After her garage rent is withdrawn, the new balance is \(-\$185\). Write an equation using \(x\) for the garage rent, and solve for \(x\).

Write and solve an equation for each number puzzle. a) Subtracting \(55\) from a number \(y\) gives \(-25\). b) Adding a number \(z\) to \(-140\) gives \(60\).

Mr. Schmidt pays a \(\$545\) bill from his checking account. Soon afterward, a credit of \(\$210\) is deposited. His balance is then exactly \(-\$185\). What was his account balance before these two transactions?

An elevator in a parking garage first travels up \(5\) levels and then down \(8\) levels. It ends on Level \(-2\). On which level did the elevator start?

During one level of a video game, Tim's score changes several times. He first earns \(120\) bonus points, then loses \(250\) points, and finally earns another \(40\) points. His score at the end of the level is \(-30\). What was Tim's score at the beginning of the level?

A withdrawal of \(\$150\) changes an account balance. After the withdrawal, the balance is \(-\$35\). Write and solve an equation to find the balance before the withdrawal.

Luke says, “When I subtract \(240\) from my number, I get \(360\).” Marie says, “When I add \(180\) to my number, I get \(800\).” Write and solve one one-step addition or subtraction equation for each person. Then determine who chose the greater number and by how much.

A kayak rental company offers two plans. Plan A: a \(\$12\) base fee plus \(\$4\) per hour. Plan B: no base fee and \(\$7\) per hour. a) Write an expression for the total cost of each plan for \(t\) hours. b) Find the cost of each plan for \(2\) hours and for \(5\) hours. Which plan is less expensive each time? c) Explain what the number \(12\) and the variable \(t\) mean in Plan A.

A storage tank contains \(V\) gallons of oil. A machine uses \(12\) gallons each day. a) Write an expression for the amount of oil remaining after \(n\) days. b) Evaluate the expression for \(V=500\) and \(n=10\). c) What does it mean in this context if the expression has a value of \(0\)? d) How does the expression change if the machine uses an unknown amount of \(x\) gallons per day?

The total cost of manufacturing a part includes material cost, labor cost, and a fixed \(\$100\) machine fee. Labor costs \(25\%\) more than materials. The total cost is \(\$1900\). Let \(x\) be the material cost. Write a one-variable equation that models all three cost components, solve it, and report the material cost.

A bus for a class field trip costs a flat \(\$450\). Zoo admission costs \(\$12\) per person. The total trip cost is \(\$774\). Let \(x\) be the number of people. Write and solve an equation for the total trip cost, then report how many people are attending.

A farm dries lavender for scented sachets. Fresh lavender loses \(65\%\) of its weight when dried. a) How many pounds of dried lavender are produced from \(14\,\text{lb}\) of fresh lavender? b) A large order requires \(7\,\text{lb}\) of dried lavender. The farm has \(18\,\text{lb}\) of fresh lavender available. Is that enough? Justify your answer.

A water tank initially contains \(B\) gallons. Pipe A adds \(a\) gallons per minute, Pipe B adds \(b\) gallons per minute, and a drain removes \(c\) gallons per minute. After \(m\) minutes, the tank contains \(W\) gallons: \( W=B+m(a+b-c). \) a) Rearrange the formula to solve for \(m\). b) The tank's capacity is \(V\). Write \(V-W\) in terms of \(B,m,a,b,c\), and explain what that expression represents. c) State the physical conditions on \(m\) under which the model is meaningful if the tank cannot contain less than \(0\) or more than \(V\) gallons.

A streaming service charges a monthly base fee of \(\$8.50\) plus \(\$2.50\) for each rented movie. a) Write an expression for the monthly cost when \(n\) movies are rented. b) Find the cost for a month with \(6\) rentals. c) A competing service has no base fee but charges \(\$4.00\) per rental. Which service costs less for \(5\) rentals? Show the calculation.

Two adjacent angles form a straight angle, so their measures add to \(180^\circ\). One angle is \(44^\circ\) greater than the other. Find both angle measures by writing and solving an equation.

Three adjacent angles lie along a straight line. The second angle is three times the first angle. The third angle equals the sum of the first two angles. Find all three angle measures by writing and solving an equation.

The sum of four consecutive integers is \(66\). Use an equation to find the four integers.

Two rectangles have the same width. The first rectangle is \(8\,\text{in.}\) high, and the second is twice as high. Their combined area is \(192\,\text{in.}^2\). Find their common width and the area of each rectangle.

A \(100\,\text{in.}\) rope is cut into three pieces. The first piece is twice as long as the second piece. The third piece is exactly \(5\,\text{in.}\) shorter than the first piece. Let \(x\) be the second piece’s length. Write one equation that models all three pieces and the total length, solve it, and report every piece length.

Three seventh-grade classes raise a total of \(\$510\) for an environmental project. Class B raises exactly two-thirds as much as Class A. Class C raises \(\$15\) less than Class B. Let \(x\) be Class A’s amount. Write one equation that models the three class amounts, solve it, and report how much each class raised.

Three sections of temporary fencing are installed at a construction site. The first section is \(4a+3b\) feet long. The second section is \(a-b+2\) feet longer than the first section. The third section is \(2a+4b-5\) feet long. For values of \(a\) and \(b\) that make all three lengths positive, write and simplify an expression for the total length of the fencing.

For values of \(a\) and \(b\) that make all original and new dimensions positive, a rectangle has length \(L=4a+3b\) and width \(W=2a+b\). A second rectangle is built with a length that is \(a-b\) greater than \(L\) and a width that is \(b\) less than \(W\). Write and simplify an expression for the perimeter of the second rectangle.

For values of \(a\) and \(b\) that make all side lengths positive and form a valid triangle, the first two sides are Side 1: \(a+4b\) Side 2: \(3a-2b+5\). The third side is \(2a+b+3\) less than the sum of the first two sides. Write and simplify an expression for the perimeter \(P\).

A school cafeteria sells sandwiches for \(\$2.00\) and bottles of apple juice for \(\$1.50\). Total sales were \(\$110.00\), and exactly twice as many sandwiches as juice bottles were sold. Let \(x\) be the number of juice bottles. Write a one-variable revenue equation, solve it, and report both quantities.

A hiking group travels a total of \(46\,\text{miles}\) over three days. On the second day, the group hikes \(1.5\) times the distance from the first day. On the third day, the group hikes \(2\,\text{miles}\) less than on the second day. Let \(x\) be the first-day distance. Write one equation for the three-day total, solve it, and report the distance hiked each day.

Three classes raise a total of \(\$720\) for a community project. Class B raises \(\$40\) more than Class A. Class C raises the same amount as Classes A and B combined. 1) Let \(x\) be Class A’s amount. Write and solve one equation that models all three contributions, then report each amount. 2) What fraction or percent of the total is raised by Class C?

A triangle has a perimeter of \(27\,\text{in.}\). The second side is \(3\,\text{in.}\) longer than the first side, and the third side is twice as long as the first side. a) Write an equation that can be used to find the first side length \(x\). b) Find all three side lengths. c) How would the equation in part a) change if the third side were \(3\,\text{in.}\) shorter than the first side instead of twice as long?

Consider the equation \(x+(x-2)+3x=43\). a) Create a number puzzle or real-world situation represented by this equation. Briefly explain what \(x\), \(x-2\), and \(3x\) mean in your situation. b) Solve the equation. Give the value of \(x\) and the values of the other two expressions.

A beverage plant uses \(4\) Type A machines and \(5\) Type B machines at the same time. Each Type A machine fills \(x\) bottles per minute, each Type B machine fills \(y\) bottles per minute, the order contains \(F\) bottles, and the filling time is \(t\) minutes. The quantities satisfy \( F=t(4x+5y). \) a) Rearrange the formula to solve explicitly for \(t\). b) Find \(t\) when \(x=25\), \(y=20\), and \(F=4000\). c) Explain why the denominator in your rearranged formula represents a rate.

A landscaping crew has \(2\) experienced workers and \(3\) assistants. Each experienced worker mows \(f\) square feet per hour, each assistant mows \(h\) square feet per hour, the lawn area is \(A\) square feet, and the work takes \(d\) hours. For the original crew, \( A=d(2f+3h). \) a) Rearrange the formula to solve for \(d\). b) One experienced worker is replaced by two assistants. Write the corresponding area equation using a new time \(d_{\text{new}}\), then rearrange it to solve for \(d_{\text{new}}\). c) Use the two solved formulas to determine when the new crew is faster than the original crew. d) Find the original time when \(f=1200\), \(h=800\), and \(A=19{,}200\).

An empty water tank holds \(V\) gallons. Pipe A adds \(x\) gallons per minute, and Pipe B adds \(5\) gallons per minute more than Pipe A. If both pipes run for \(t\) minutes, then \( V=t(2x+5). \) a) Rearrange the formula to solve for \(t\). b) Find \(t\) when \(V=750\) and \(x=35\). c) Rearrange the same formula to solve for \(x\) in terms of \(V\) and \(t\).

Boxes are stacked on a pallet in \(r\) rows with \(k\) boxes per row. Each box weighs \(m\) pounds, and the empty pallet weighs \(50\) pounds. If the loaded pallet weighs \(M\) tons, then \( M=\frac{rkm+50}{2000}. \) a) Rearrange the formula to solve for the box weight \(m\). b) Find \(m\) when \(M=0.905\), \(r=5\), and \(k=8\). c) Explain the purpose of the factor \(2000\) in the original formula.

A sports club buys equipment costing \(\$K\). A city grant contributes \(\$Z\), and the remaining cost is divided equally among \(s\) active members. If each member pays \(P\) dollars, then \( P=\frac{K-Z}{s}. \) a) Rearrange the formula to solve for \(Z\). b) Find the grant \(Z\) when \(K=540\), \(s=15\), and \(P=28\). c) Explain why increasing \(Z\) lowers \(P\) when \(K\) and \(s\) stay fixed.

Consider the formula \(x=\frac{S-b}{k}\). a) Create a real-world problem that can be modeled by this formula. Explain what \(S\), \(b\), and \(k\) represent in your problem. b) Find \(x\) when \(S=85\), \(b=13\), and \(k=12\).

A cycling group plans a three-day, \(180\,\text{mile}\) trip. The second-day distance will be twice the first-day distance. The third-day distance will be \(15\,\text{miles}\) shorter than the second-day distance. a) Write an equation to find the first-day distance \(x\). b) Find the distance for each day. c) Without solving again, explain how the equation from part a would change if the third-day distance were instead \(10\,\text{miles}\) longer than the first-day distance.

A \(750\,\text{ft}^2\) school garden is divided into a vegetable garden, a flower garden, and a lawn. The lawn has the same area as the vegetable and flower gardens combined. The vegetable garden has two-thirds the area of the flower garden. Find the area of each section.

A teacher buys \(6\) packs of highlighters and \(12\) pencils for \(\$54.00\). One pack of highlighters costs four times as much as one pencil. a) Use an equation to find the price of one pencil and one pack of highlighters. b) A student claims, “If a pack of highlighters cost only three times as much as a pencil, the savings would buy exactly \(6\) more pencils.” Determine whether the claim is correct.

A workshop uses \(8\) identical LED floodlights and \(2\) identical heaters. Together, they draw \(3600\,\text{W}\). Each heater draws five times as much power as one floodlight. a) Let \(s\) be the power draw of one floodlight. Write and solve an equation for the original system, then report the power draw of one floodlight and one heater. b) The heaters are upgraded so each uses \(20\%\) less power. What is the new total power draw of all \(8\) floodlights and \(2\) heaters?

A concentrated vinegar solution is \(25\%\) acetic acid by volume. A recipe requires vinegar that is \(5\%\) acetic acid. Assume the volumes add. a) How many milliliters of water must be added to \(100\,\text{mL}\) of the concentrate to make a \(5\%\) solution? b) A cook wants to make exactly \(1\,\text{L}\) of the \(5\%\) solution. How many milliliters of concentrate and how many milliliters of water are needed?

A fruit-juice mixture contains \(x\) liters of concentrate and \(y\) liters of water. The concentrate contains \(s\) grams of sugar per liter. If the final mixture has sugar concentration \(q\) grams per liter, then \( q=\frac{xs}{x+y}. \) a) Rearrange the formula to solve for \(s\). b) Find \(s\) when \(q=30\), \(x=0.4\), and \(y=1.6\). c) Explain why the factor \(x+y\) appears in your rearranged formula.

A smartphone battery is at \(80\%\) at the start of an observation. During intensive use, its charge decreases by \(12\) percentage points per hour. Write an inequality and find the times \(t\) when the charge is below \(20\%\). Consider only times up to complete discharge.

For a school celebration, one-third of the budget is spent on renting the space, two-fifths is spent on food, and one-sixth is spent on drinks. Exactly \(\$30\) remains for decorations. What is the total budget?

In a school library, two-fifths of the items are novels and one-fourth are nonfiction books. Exactly half of the remaining items are children’s books. The final \(105\) items are magazines. How many items are in the library altogether?

Maya is reading a new book. On the first day, she reads exactly one-third of the pages. On the second day, she reads one-fourth of the total number of pages. She then has exactly \(125\) pages left. Write an equation and find the total number of pages in the book.

A school project budget assigns \(\frac{3}{8}\) to materials, \(\frac{1}{5}\) to transportation, \(\frac{1}{10}\) to food, and \(\frac{1}{4}\) to equipment rental. The remaining \(\$15\) is set aside as an emergency fund. Find the total budget and the amount budgeted for materials.

Jordan saved money for a project. First, Jordan spends \(\frac{1}{4}\) of the money on reference books. Then Jordan spends \(\frac{2}{5}\) of the remaining money on supplies. After both purchases, \(\$9.00\) remains. Find the original amount and the amounts spent on books and supplies.

A water tank initially contains \(W\) gallons. A leak drains water at a constant rate of \(k\) gallons per hour. If \(A\) is the amount of water remaining after \(h\) hours, then \( A=W-kh. \) a) Rearrange the formula to solve for \(h\). b) Find \(A\) when \(W=500\), \(k=12\), and \(h=8\). c) Use your rearranged formula to give the empty-tank time in terms of \(W\) and \(k\). d) State the physically meaningful interval of \(h\) while the tank is draining.

Two rectangular beds in a school garden have the same width. The first bed is \(45\,\text{ft}\) long, and the second is \(60\,\text{ft}\) long. The second bed has \(375\,\text{ft}^2\) more area than the first. Find their common width.

Two trains leave the same station at the same time and travel in opposite directions. After \(1.5\) hours, the trains are \(150\) miles apart. The slower train travels at exactly \(\frac{2}{3}\) the speed of the faster train. Let \(v\) be the faster train’s speed. Write one equation using distance equals rate times time, solve it, and report both speeds.

Two pumps fill a tank. Pump A moves \(x\) gallons per minute and Pump B moves \(y\) gallons per minute. After both pumps have run for \(t\) minutes, the tank still needs \(R\) gallons to reach its capacity \(V\). The quantities satisfy \( V=(x+y)t+R. \) a) Rearrange the formula to solve for \(t\). b) Find \(t\) when \(V=630\), \(x=15\), \(y=20\), and \(R=210\). c) Rearrange the original formula to solve for \(R\).

The diagram shows the composition of an urn. One ball is selected at random. a) Find the probability of selecting a blue ball. Write the probability as a fraction in simplest form and as a percent. b) How many gray balls must be added so that the probability of selecting a blue ball is exactly \(25\%\)?

The number line shows a temperature change from the previous evening to the morning low. Find the previous evening's temperature.

The diagram shows quadrilateral \(ABCD\) with a circle tangent to all four sides. a) State the tangent-segment fact about two tangent segments drawn from the same external point. b) Use that fact to derive a relationship among the four side lengths of this quadrilateral. c) Use the side lengths shown in the diagram to find \(d\).

A recreation center charges a \(\$25\) room fee plus \(\$7\) per participant. Situation A: One group is charged exactly \(\$74\). Situation B: Another group has a budget of no more than \(\$74\). For each situation, decide whether an equation or an inequality is appropriate, write the model, solve it, and explain the difference in the meanings of the solutions.

A fundraiser already has \(\$32\) and earns \(\$6\) for each batch of cards sold. The goal is to raise at least \(\$80\). A student writes \(32+6b\le80\). a) Explain the modeling error. b) Write the correct inequality. c) Solve it and interpret the answer.

A warehouse lifting system can support a maximum combined weight of \(1200\,\text{kg}\), including the platform itself and its cargo. During one job, the cargo changes as follows: 1. Load \(350.5\,\text{kg}\). 2. Unload \(120\frac{1}{4}\,\text{kg}\). 3. Load \(480.75\,\text{kg}\). 4. Unload \(200\,\text{kg}\). 5. Load \(150.5\,\text{kg}\). a) How much cargo is on the platform after step 5? b) What is the greatest possible platform weight that keeps the combined weight at or below \(1200\,\text{kg}\) at every step?

Hikers use this rule to estimate travel time: “Allow one hour for every \(3\) miles of horizontal distance. Add one more hour for every \(2000\) feet of elevation gain.” a) Write an equation for the total time \(T\), in hours. Let \(s\) be the horizontal distance in miles and \(h\) be the elevation gain in feet. b) Estimate the time for a hike of \(7.5\) miles with \(3000\) feet of elevation gain. c) A group claims, “If we double the distance but keep the elevation gain the same, the estimated time also doubles.” Use your equation to evaluate the claim.

Fence posts are installed with equal spacing. The gap from the first post to the beginning of the fence line and the gap from the last post to the end are the same as the gaps between posts. With \(12\,\text{ft}\) gaps, a certain number of posts is needed. With \(7.5\,\text{ft}\) gaps, \(3\) more posts are needed. Let \(x\) be the number of posts in the first arrangement. Write an equation with \(x\) on both sides by equating the two expressions for the same fence length. Solve that equation algebraically, then find the fence length.

A temporary construction fence uses \(12\,\text{ft}\) panels and connecting posts that are \(4\,\text{in.}\) wide. A fence section begins and ends with a post, with one panel between each pair of posts. a) Write a formula for the total length \(L\), in feet, of a fence with \(n\) panels. b) How many panels are needed to cover at least \(160\,\text{ft}\)? c) How many posts are needed for that number of panels? Explain the relationship.

Noah and Maya each have exactly \(48\,\text{ft}\) of wire for a wireframe model. Noah builds a cube. Maya builds a rectangular prism with a square base whose height is twice a base edge. a) Let \(s\) be Noah's cube edge length. Write and solve an equation for \(s\). b) Let \(a\) be Maya's base-edge length. Write and solve an equation for \(a\) using the wire lengths of all \(12\) prism edges. c) Use your two equations to find the difference between the model heights.

A stack of freshly cut wood weighs \(120\,\text{lb}\) and is \(85\%\) water. After drying, the wood is \(80\%\) water. a) Find the new total weight of the wood. b) Explain why the total weight decreases substantially even though the water percentage decreases by only \(5\) percentage points.

A class sells waffles at a school carnival. Fixed costs for equipment, batter, and supplies are \(\$42.00\). Each waffle also costs \(\$0.35\) in toppings and is sold for \(\$1.50\). a) First find the net amount from each waffle that is available to cover fixed costs and profit. Use that net amount to write and solve an inequality for avoiding a loss. Then give the least whole number of waffles the class must sell. b) Write and solve an inequality for earning a profit of at least \(\$120.00\). Then give the least whole number of waffles the class must sell.

Isabelle has \(40\) counters divided among three piles: left \(L\), middle \(M\), and right \(R\). She does not know the amount in each pile. The left pile has at least \(7\) counters, and the right pile has at least \(5\). She moves \(7\) counters from \(L\) to \(M\), then moves \(5\) counters from \(R\) to \(M\). After the moves, a student reports that the left and right piles contain \(15\) counters altogether. Isabelle immediately says, “Then the middle pile started with exactly \(13\) counters.” Use equations with variables to show that Isabelle is correct.

Class A has \(25\) students, including \(12\) boys. Class B has \(20\) students. If \(4\) new boys joined Class B, boys would make up exactly \(50\%\) of that class. Find the original percent of students in Class B who were boys. Which class originally had the greater percent of boys?

In a class of \(40\) students, \(60\%\) regularly play on organized sports teams. Of those team athletes, \(25\%\) play on a soccer team. a) How many students play on a soccer team? b) Some students who do not play organized sports hike regularly. Let \(h\) be the number of these hikers. The team athletes and hikers together must make up more than \(80\%\) of the class. Write and solve an inequality for \(h\), then give the least whole number of hikers that satisfies the condition. c) If the number of hikers is the least value from part b), how many students are neither team athletes nor hikers?

A box contains \(40\) marbles. Of the marbles, \(25\%\) are blue and the rest are white. How many blue marbles must be added so that the probability of drawing a blue marble is exactly \(40\%\)? Do not change the number of white marbles. Justify your reasoning.

A new candle is \(10\,\text{in.}\) tall. Once lit, it burns down at a constant rate of \(0.4\,\text{in.}\) per hour. For what burning times \(h\) is the candle shorter than \(4\,\text{in.}\)? Consider only times up to the moment when the candle burns out completely.

A rancher is asked how many sheep are in the flock and replies, “If I tripled my flock, sold one-fourth of the original number of sheep, and then received \(2\) more sheep, I would have exactly \(68\) sheep.” a) Write and solve an equation to find the original number of sheep. b) An apprentice claims the statement must be impossible because calculations involving fourths can produce fractional sheep. Determine whether that concern applies here.

Two rectangular flower beds have the same length. Bed A is \(3\,\text{ft}\) wide, and Bed B is \(5\,\text{ft}\) wide. Bed B has \(14\,\text{ft}^2\) more area than Bed A. a) Use an equation to find the common length. b) Bed C has the same length and a width equal to the sum of the other two widths, \(8\,\text{ft}\). Find the area of Bed C. c) Explain why the area of Bed C equals the sum of the areas of Beds A and B.

A store sells large wooden crates for \(\$12.00\) and small crates for \(\$4.50\). A customer spends \(\$153.00\) and buys three times as many small crates as large crates. a) Use an equation to find how many crates of each size the customer bought. b) The customer bought \(24\) crates in all. What would the total cost have been if the customer had instead bought twice as many large crates as small crates?

A wood-pellet bin holds \(M\) pounds when full. Filling it completely costs \(\$S\), and the pellets have a constant price per pound. A heating system uses pellets worth \(\$k\) each day. After \(t\) days, \(R\) pounds remain. These quantities satisfy \( R=M-\frac{kMt}{S}. \) a) Rearrange the formula to solve for \(t\). b) Find \(t\) when \(M=3000\), \(S=1200\), \(k=10\), and \(R=2250\). c) Explain why \(S\) appears in the numerator of your solved formula for \(t\).

A youth group with \(n\) members rents a bus costing \(\$B\). The cost is shared equally. Then \(m\) more people join, the bus cost rises by \(\$s\), and each person's savings relative to the original per-person cost is \(C\) dollars. These quantities satisfy \( C=\frac{B}{n}-\frac{B+s}{n+m}. \) a) Rearrange the formula to solve for \(s\). b) Use your rearranged formula to find \(s\) when \(B=400\), \(n=20\), \(m=5\), and \(C=2\). c) Explain what your value of \(s\) means in the situation.

A chemist has \(200\,\text{g}\) of a \(10\%\) salt solution. The chemist adds \(x\) grams of a \(40\%\) salt solution to make a mixture that is exactly \(28\%\) salt. a) Write an equation with \(x\) on both sides that equates the amount of salt before and after mixing. b) Solve the equation algebraically. Show at least two equivalent-equation steps after writing the model. c) Interpret the solution.

A cyclist travels for \(t_1\) hours at \(v_1\) miles per hour and then for \(t_2\) hours at \(v_2\) miles per hour. The average speed over the whole trip is \( v_{\text{avg}}=\frac{v_1t_1+v_2t_2}{t_1+t_2}. \) a) Rearrange the formula to solve for \(v_2\). b) Find \(v_2\) when \(v_{\text{avg}}=14\), \(v_1=10\), \(t_1=2\), and \(t_2=3\). c) Check your result in the original average-speed formula.

A school garden is divided into different areas. Flower beds cover \(\frac{2}{7}\) of the garden, and vegetables cover exactly one-half of the garden. The rest is a paved path. The path has an area that is exactly \(30\,\text{ft}^2\) less than the flower-bed area. a) What fraction of the garden is the path? b) What is the total area of the garden?

At an apple-processing facility, \(5\%\) of a delivery is removed because the apples are damaged. The remaining apples are pressed, and the juice produced weighs \(75\%\) as much as the apples that are pressed. How many pounds of apples must be delivered to produce \(570\,\text{lb}\) of juice?

A composting facility analyzes the organic waste delivered in one day. The pie chart shows the composition of the waste. The facility found exactly \(12\) tons of contaminants, such as plastic or metal, in the deliveries. How many tons of food scraps and yard waste were delivered altogether?

A theater group buys \(6\) identical tickets. Each ticket costs \(x\) dollars plus a \(\$2.75\) booking fee. The total charge is \(\$76.50\). a) Write an equation that models the purchase. b) Find the ticket price \(x\) before the booking fee. c) If the ticket price before the fee were \(\$1.50\) higher, how much would the total charge increase? Explain using the structure of your model.