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One Variable Equations
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These problems can be solved by using a single variable and generating an equation. Students may prefer to do some of the problems by using a system of equations if they are familiar with that
approach. Problems cover a variety of topics, among them distance/rate/time, work, and some basic geometry.
- Related Resources
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Interactive resources from our Math Tools project:
Algebra: One Variable Equations
The closest match in our Ask Dr. Math archives:
Middle School: Equations
NCTM Standards:
Algebra Standard for Grades 9-12
Access to these problems requires a Membership.
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'Ace' Trotter
- Terry Trotter
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Algebra, difficulty level 1. A problem from the sports world regarding free-throw percentages.
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Ant Race
- Terry Trotter
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Algebra, difficulty level 1. Two ants race to a piece of food. Your task is to figure out how far apart they were when the race started.
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Approval Method of Voting
- Ethel Breuche
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Algebra, difficulty level 3. In the country of Algpoweria, eligible voters can cast their votes for two candidates. Based on the clues given, find how many votes each candidate received and the number of eligible voters.
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Are We There Yet?
- Terry Trotter
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Algebra, difficulty level 1. Given what fractions of a car trip have been completed at two different times, can you predict when the trip will end?
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Are You Ready for Some Football?
- Steve Risberg
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Algebra, difficulty level 2. The Algenauts football team racked up 229 yards rushing in their first
game. Given descriptions of how many yards each of the players ran
for in relation to each other, determine the exact yardage each one had.
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An "Average" Problem
- Terry Trotter
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Algebra, difficulty level 2. Three test grades are removed from a set of scores. You are given the new set's average and a ratio involving the scores removed. What scores have been removed?
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Bowling Buddies
- Steve Risberg
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Algebra, difficulty level 2. Four friends go bowling and have a small contest. From the clues
given about their scores, can you determine who won the contest?
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Boxes of Cereal
- Steve Weimar
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Algebra, difficulty level 2. Lily and Mikey consume their boxes of cereal at different rates. Use
the hints to determine how much cereal is in each child's box.
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The Box Supper
- Terry Trotter
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Algebra, difficulty level 3. Find the amount of money that Judy Ann's "beau" paid at the box supper auction for her box of delicious food.
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Bugs Bunny's Money
- Terry Trotter
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Algebra, difficulty level 2. Find how much Mr. Bugs Bunny is carrying in a money bag, given clues such as the ratio of coins that are pennies and the percentage of coins that are dimes.
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Calling Long Distance
- Terry Trotter
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Algebra, difficulty level 1. Use simple equations to compare two telephone calling plans.
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Carlita's CD Collection
- Terry Trotter, Steve Risberg
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Algebra, difficulty level 2. Carlita describes how many CD's are in her collection in a rather
mathematical way. Can you figure out how many there are?
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Cherokee Creek Checkers Club
- Terry Trotter
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Algebra, difficulty level 1. Until one month ago, the Checkers Club had won 40 of 90 matches so far this year. But last
month they won 3 of every 5 games they played. How many games did
they play last month if their overall winning percentage is now 52%?
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A Day at the Fairs
- Terry Trotter via Kraitchik & Chuquet
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Algebra, difficulty level 2. A merchant went selling and buying at three city fairs. Did he make a profit or take a loss?
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Dinner at Pepe's
- Steve Risberg
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Algebra, difficulty level 1. Three friends share a pizza and sodas. Given information about the
portion of the bill each one paid, determine the total cost of
the meal.
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Donald Duck's Trip
- Terry Trotter
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Algebra, difficulty level 2. What fractional part of Donald's trip had he traveled by 4:00 P.M.?
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Driving to Work
- Annie Fetter
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Algebra, difficulty level 2. Tracey drives to work at 40 mph and is 1 minute late. If she leaves at the same time and drives 45 mph, she'll be 1 minute early. How far does Tracey live from work?
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Examining an Equation
- Steve Risberg
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Algebra, difficulty level 2. Find integer solutions to an equation in two variables.
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Filling the Aquarium
- Terry Trotter
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Algebra, difficulty level 2. A workman at a zoo has to clean an aquarium, but forgets to do an important task, thus causing a problem (for us).
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Flatland Soldiers
- Terry Trotter
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Algebra, difficulty level 1. Find the size of the vertex angle of a soldier in the world of Flatland, a famous book by Edwin A. Abbott.
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Flying Off the Shelves
- Steve Risberg
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Algebra, difficulty level 1. The new soundtrack album from the movie Ray is selling like hotcakes!
Given some information about the sales over three days at Tip Top
Tunes, can you determine how many CDs sold per day?
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Fording the River
- Lisa and Judy Gardner
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Algebra, difficulty level 2. A terrible accident occurred just when the hands of a watch were in a special position. What time was it?
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Four Men and Some Money
- Daboll
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Algebra, difficulty level 1. Four men (A, B, C, & D) are splitting up a quantity of money ($100), and need your help to do it.
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Grandpa Jones' Age
- Terry Trotter
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Algebra, difficulty level 2. Little George asks how old his grandpa is, and needs your help to find out the answer.
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The Great One
- Terry Trotter
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Algebra, difficulty level 2. Indiana Jones found a math problem about the life span of "The Great One" engraved on an old tombstone. How old was the Great One when he died?
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Holiday Dance
- Steve Risberg
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Algebra, difficulty level 2. Two bands are being considered for the holiday dance at Rahkenrole
High School. Each band charges a base fee and an hourly rate. Help
the Student Council compare the bands and find the best deal.
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Jack and Jill Go Jogging
- Terry Trotter
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Algebra, difficulty level 2. Two famous Mother Goose characters take up aerobics, running at some numerical speeds while exercising. How fast can each one jog?
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Jimmy's Hits
- Terry Trotter & Annie Fetter
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Algebra, difficulty level 2. Given information about how many doubles, homers, and singles, figure out how many total hits Jimmy had during his record-breaking season.
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Judy and Jen's Garden Problem
- Terry Trotter
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Algebra, difficulty level 2. Judy and her daughter Jen make up math problems while working in their vegetable garden.
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Let It Snow!
- Steve Risberg
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Algebra, difficulty level 3. Use the given statements to determine how many clear days there were
in a recent span of snowy days.
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Lettuce for Sale
- Ethel Breuche
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Algebra, difficulty level 1. Using a linear equation, determine the number of heads of lettuce sold on Monday.
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Louis And Boris Get Even
- Ethel Breuche
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Algebra, difficulty level 2. After taking their respective hormone pills, Louis is growing and Boris is shrinking. How long will it take before they become the same height?
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Lucie and Daryl's Bike Ride
- Terry Trotter
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Algebra, difficulty level 2. A sister and brother who ride bikes provide us with a math challenge concerning distance, rate, and time.
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Make the Save!
- Steve Risberg
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Algebra, difficulty level 1. A hockey goalie has saved 112 out of 131 shots so far this season.
How many consecutive shots does she need to save to raise her save
percentage to .900?
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Marathon Mitch
- Terry Trotter
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Algebra, difficulty level 2. Mitch runs a road race and increases speed twice during the race.
Given his total finishing time, determine his standard running rate.
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Missing Angle Mystery
- Terry Trotter
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Algebra, difficulty level 2. Help Detective Sam Spade solve a robbery of valuable polygons from Mrs. Hempstead's famous collection.
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Mr. Green's Birthday
- Ethel Breuche
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Algebra, difficulty level 1. Mr. Green has a birthday but will not tell his age directly. Can you figure out his age based on the clues?
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My Bag of Marbles
- Terry Trotter
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Algebra, difficulty level 2. Find the probability of drawing a blue marble after additional marbles have been placed in a bag.
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Number, Please!
- Terry Trotter
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Algebra, difficulty level 4. Given two telephone calling plans, find the point at which the monthly bills would be the same.
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Painting Partners
- Terry Trotter
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Algebra, difficulty level 1. Two house painters combine their skills and talents to make short order of a painting job.
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Partners in Pizza
- Terry Trotter
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Algebra, difficulty level 2. Three friends share the cost of a pizza. How much does each contribute?
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A Phone-y Deal?
- Steve Risberg
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Algebra, difficulty level 2. Compare two cell phone pricing plans to determine for how many minutes
of calls per month each one is the better deal.
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Pressed for Time
- Steve Risberg
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Algebra, difficulty level 2. Due to a late breaking story, a newspaper must use both presses to get
an edition printed in time. Find how long it takes to print a normal
edition using only the newer press.
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Profits and Lots
- Terry Trotter
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Algebra, difficulty level 1. A person buys some land, keeps some for himself, divides the rest into lots, and sells them, thus making a profit.
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Ration Ratios
- Steve Risberg
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Algebra, difficulty level 3. Determine the ratio of vegetarian meals to vegetarians at the luncheon.
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Sam and Teri's Bank Accounts
- Terry Trotter
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Algebra, difficulty level 1. Sam and Teri have bank accounts. Sam always withdraws money; Teri always saves it. When will they have equal balances, or will they?
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A Sequence of Numbers
- Asher Walkover
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Algebra, difficulty level 2. Find the second number in a Fibonacci-like series.
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Ships Ahoy!
- Terry Trotter
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Algebra, difficulty level 1. One-third of a fleet was captured, one-sixth was sunk, and two ships were destroyed by fire. One-seventh of the surviving ships were later lost in a storm. The 24 remaining ships sailed home. How many ships were there before the engagement?
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Skeeball Showdown
- Steve Risberg
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Algebra, difficulty level 3. Kira and Daquan each play a game of Skeeball to see who gets the
higher score. Given descriptions of their results in relation to each
other, determine the final score of their match and how many balls
each player rolled.
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Splash!
- Steve Risberg
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Algebra, difficulty level 1. How many throws did it take each student to sink Mr. Rashanul in the
dunking booth at the carnival?
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A Surprising Sequence
- Steve Risberg
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Algebra, difficulty level 2. In a Fibonacci sequence, the first number is 1 and the tenth number is
111. Find the second number.
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Susita's Savings Account
- Terry Trotter
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Algebra, difficulty level 2. Find the original balance in Susita's savings account.
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Swapping Marbles
- Lillian Ray, Steve Risberg, and Steve Weimar
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Algebra, difficulty level 3. Three kids swap marbles in an interesting way. After the swapping, which of the three had gained the most marbles? How many did they gain?
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Teeter-Toddlers
- Steve Risberg
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Algebra, difficulty level 1. Two kids of different weights balance on a seesaw. How far from the
center is each one sitting?
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Tele-Plus
- Terry Trotter
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Algebra, difficulty level 1. Find the cost of one phone that is part of a three-in-one communications package deal.
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That's Some Sequence!
- Terry Trotter
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Algebra, difficulty level 4. Find the product of 3 terms of an unusual sequence.
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Tiffany's Triumphs
- Terry Trotter
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Algebra, difficulty level 1. Given the number of games that Tiffany has played so far and the number that she has won, how many games would she have to win in a row to get her winning percentage up to 60%?
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Time Trial Triumphs
- Steve Risberg, Terry Trotter, and Annie Fetter
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Algebra, difficulty level 2. At 9:05 a.m. Ernesto starts riding at a speed of 440 yards per
minute. At 9:20 a.m. Johnny starts riding, but he's going 20% faster
than his little brother. At what time, to the nearest minute, will
Johnny catch Ernesto?
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To B or Not To B
- Steve Risberg
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Algebra, difficulty level 2. Given a letter pattern, find expressions for the number of B's and the
ratio of B's to total letters in the nth term.
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Touchdown Toddlers
- Terry Trotter
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Algebra, difficulty level 2. Two little boys run an unusual race on a professional football field, and everybody comes out a winner!
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Tribonacci
- Terry Trotter
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Algebra, difficulty level 3. Find the second and third terms of a "Tribonacci" sequence.
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Tribonacci Tribulation
- Steve Risberg
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Algebra, difficulty level 2. Given data describing a tribonacci sequence, find a missing number in
the sequence.
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Two by Canoe
- Annie Fetter
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Algebra, difficulty level 3. Figure out what time Tashawn and Forrest need to start back downstream after paddling upstream for two days if they need to meet Forrest's mother at 4 pm on the third day.
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Two Ladders
- Terry Trotter
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Algebra, difficulty level 2. Find the lengths of two ladders, then find their "differences."
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The Unhappy Carpenter
- Ethel Breuche
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Algebra, difficulty level 1. A carpenter agrees to be paid a certain amount of money when she works, and to repay money on the days she doesn't work. Find the number of days she worked if she earned $77.00.
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Up the Down Elevator
- Ethel Breuche
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Algebra, difficulty level 2. Juanita has to use a freight elevator to deliver packages but can use the passenger elevator to return. What is her average speed for the entire trip?
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A Variable Triangle
- Steve Risberg
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Algebra, difficulty level 4. Given a triangle with side lengths of x + 2, x + 3, and 11, what can
we conclude about x?
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The Weight of Bart's Brick
- Terry Trotter
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Algebra, difficulty level 1. Bart was asked by his construction foreman about the weight of a brick. Bart's answer befuddled his boss. (But not you.)
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What's the Difference?
- Steve Risberg
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Algebra, difficulty level 3. Does multiplying a number by four always give you a greater number
than you started with?
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Wreck Tangle
- Steve Risberg
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Algebra, difficulty level 2. When the width of a given rectangle is tripled and the length is
doubled, the perimeter increases by 92 cm. Find the area of the
original rectangle.
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