Priya's phone plan costs $\$20$ a month plus $\$5$ for each extra gigabyte of data. This month her bill was $\$50$. How many extra gigabytes did she use? You could guess and check, but there is a faster and surer way: turn the story into an equation. Solving the equation is the mechanical part. The skill that matters is setting it up.
The four-step method
Define the unknown. Write down exactly what your letter means, with units: "Let $t$ = the price of one ticket in dollars."
Translate each fact in the story into math, using your letter.
Set up an equation from the sentence that says two things are equal, then solve it.
Check the answer in the original story, not just in your equation, and answer with a full sentence and units.
Translating is the hard part. Here is a small dictionary.
In words
In algebra
a number
$x$
3 more than a number
$x + 3$
5 less than a number
$x - 5$
twice a number
$2x$
half of a number
$\tfrac{x}{2}$
two parts add up to 20; one part is $x$
the other is $20 - x$
"is", "equals", "would be", "gives"
$=$
The last row of the dictionary is a big time saver. If two amounts add to a known total, call one of them $x$ and the other one "total minus $x$". That way you only need one letter.
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If we add $12$ to three times a number, we get $48$ as a result. What is the number?
If we add $12$ to three times a number, we get $48$ as a result. What is the number?
Emma bought $3$ identical t-shirts for the track team and a pair of sweatpants that cost $15 \text{ \$}$. If she spent $60 \text{ \$}$ in total, how much did each…
Emma bought $3$ identical t-shirts for the track team and a pair of sweatpants that cost $15 \text{ \$}$. If she spent $60 \text{ \$}$ in total, how much did each t-shirt cost?
Consecutive integers follow one another: if the first is $x$, the next two are $x + 1$ and $x + 2$. Consecutive even (or odd) integers go up by 2: $x,\ x + 2,\ x + 4$.
Digit puzzles: a two-digit number with tens digit $t$ and units digit $u$ has value $10t + u$. The number 47 is $10 \cdot 4 + 7$, not $4 + 7$. Reversing the digits gives $10u + t$.
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Find three consecutive integers whose sum is equal to $54$.
Find three consecutive integers whose sum is equal to $54$.
A number has two digits that add up to $12$. If we reverse the order of its digits, we obtain a new number that is $18$ units greater than the original number. Find the…
A number has two digits that add up to $12$. If we reverse the order of its digits, we obtain a new number that is $18$ units greater than the original number. Find the original number using an equation with a single unknown (Hint: remember that a two-digit number can be written as $10x + y$).
Age problems have one golden rule: time passes the same for everyone. If the story says "in 8 years" or "5 years ago", add or subtract the same number from every person's age. A small table keeps you organized.
Now
In 8 years
Younger sibling
$x$
$x + 8$
Older sibling
$2x$
$2x + 8$
Another useful fact: if you add $t$ years to two people, the sum of their ages goes up by $2t$.
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Alexander is $4$ years older than his sister Lucy. If the sum of their ages is $26$ years, how old is each sibling?
Alexander is $4$ years older than his sister Lucy. If the sum of their ages is $26$ years, how old is each sibling?
A mother's age is currently three times that of her son. $6$ years ago, the sum of their ages was $40$ years. How old are the son and the mother currently?
A mother's age is currently three times that of her son. $6$ years ago, the sum of their ages was $40$ years. How old are the son and the mother currently?
A father's age today is three times that of his son. $6$ years ago, the father's age was five times the age his son was at that time. How many years must pass for the…
A father's age today is three times that of his son. $6$ years ago, the father's age was five times the age his son was at that time. How many years must pass for the sum of the father's and son's ages to be exactly $100$ years?
Comparing amounts: followers, prices, "would be equal"
Many stories compare two amounts: "3 times as many", "$\$12$ less than", "half price". Pick the amount everything else is built from and call it $x$. Then write the other amount using $x$.
When a story says that after some changes two amounts "would be equal", write each changed amount and put an equals sign between them.
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In a video game store, they have sold twice as many handheld consoles as home consoles. If they have sold $18$ consoles in total, how many of each type have they sold?
In a video game store, they have sold twice as many handheld consoles as home consoles. If they have sold $18$ consoles in total, how many of each type have they sold?
On a social network, Ashley's number of followers is three times that of her brother Ethan. If Ashley lost $150$ followers and Ethan gained $50$, both would have exactly…
On a social network, Ashley's number of followers is three times that of her brother Ethan. If Ashley lost $150$ followers and Ethan gained $50$, both would have exactly the same amount. How many followers does each one currently have?
Marcus bought a video game and some headphones for his console. The headphones cost him $\$12…
Marcus bought a video game and some headphones for his console. The headphones cost him $\$12$ less than the video game. If the video game were reduced to half its original price, Marcus would have paid $\$48$ for both items together. What is the original price of each item?
For geometry stories, draw the figure and label the sides with $x$. Then use the perimeter: add up all the sides.
Rectangle: $P = 2\ell + 2w$
Triangle: add the three sides. In an isosceles triangle, two sides are equal.
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We want to put a wooden frame on a rectangular photograph. We know that the length of the photograph is $5 \text{ cm}$ more than the width. If we needed a strip of…
We want to put a wooden frame on a rectangular photograph. We know that the length of the photograph is $5 \text{ cm}$ more than the width. If we needed a strip of $70 \text{ cm}$ total length (perimeter) to frame it, what are the dimensions of the photograph?
A farmer wants to fence a triangular plot for his sheep. Knowing that it is an isosceles triangle, that the unequal side is $5$ feet shorter than each of the equal sides…
A farmer wants to fence a triangular plot for his sheep. Knowing that it is an isosceles triangle, that the unequal side is $5$ feet shorter than each of the equal sides and that $40$ feet of fencing were needed in total, how long is each side of the plot?
Perimeter adds lengths. Area multiplies them. Area stories often look scary, but notice that the $x^2$ parts usually cancel, and what is left is a one-step or two-step equation.
Idea 1: look only at what changed. If you make a rectangle taller, the new area minus the old area is just the strip you added.
Idea 2: a path or deck around a rectangle. The path adds its width on both sides, so each outer dimension grows by twice the path width. Split the path into four rectangles (two long strips and two end strips that include the corners) and add their areas.
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A rectangular tennis court has a perimeter of $60$ feet. Knowing that the length of the court is $6$ feet more than the width, calculate the dimensions (length and…
A rectangular tennis court has a perimeter of $60$ feet. Knowing that the length of the court is $6$ feet more than the width, calculate the dimensions (length and width) of the court.
Calculate the dimensions of a rectangle knowing that its base is $3$ in greater than its height and that, if we increase the height by $2$ in, the area of the new…
Calculate the dimensions of a rectangle knowing that its base is $3$ in greater than its height and that, if we increase the height by $2$ in, the area of the new rectangle increases by $16$ $cm^2$.
A rectangular garden is to be designed surrounded by a tiled path of a constant width of $2$ feet. The length of the garden is twice its width. If we know that the total…
A rectangular garden is to be designed surrounded by a tiled path of a constant width of $2$ feet. The length of the garden is twice its width. If we know that the total area occupied by the tiled path is $112$ ft$^2$, what are the dimensions of the garden (width and length)?
"20 heads and 64 legs", "24 vehicles and 64 wheels": these stories have two kinds of things with different values. Use the total-minus-$x$ trick. If there are 15 vehicles in all and $x$ of them are cars, then $15 - x$ are bikes. Each kind contributes (number of items) $\times$ (value per item).
Sometimes the last question asks you to compare. For example, after finding the numbers you may compute a cost and see whether it stays within a budget. "Does not exceed" means $\le$. If the budget is already met, the answer to "how many must change?" can be none, and $0$ is a legitimate answer.
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A shepherd has a mixture of sheep and chickens in his pen. If we count the heads there is a total of $20$, and if we count the legs there are $64$. How many sheep and…
A shepherd has a mixture of sheep and chickens in his pen. If we count the heads there is a total of $20$, and if we count the legs there are $64$. How many sheep and how many chickens are there in the pen? (Remember that sheep have $4$ legs and chickens $2$).
A courier company uses vans and motorcycles. The vans have $4$ wheels and the motorcycles $2$. In total there are $24$ drivers (one per vehicle) and $64$ wheels. Due to…
A courier company uses vans and motorcycles. The vans have $4$ wheels and the motorcycles $2$. In total there are $24$ drivers (one per vehicle) and $64$ wheels. Due to logistical problems, the daily maintenance cost is $\$15$ per van and $\$6$ per motorcycle. If the total daily maintenance budget is $\$250$, how many van drivers should switch to driving motorcycles so as not to exceed the budget without changing the total number of vehicles?
When you mix two ingredients, two different things add up:
the amounts (pounds, mL), and
the money (price per unit $\times$ amount).
Use total-minus-$x$ for the amounts. Then write one money equation: (money from ingredient 1) + (money from ingredient 2) = (price of the mix) $\times$ (total amount).
Matching units. If prices are per liter but amounts are in milliliters, convert first so they match: $\$120$ per liter is $\$0.12$ per mL. Then clear decimals by multiplying both sides by a power of 10.
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In a coffee shop, they mixed Colombian coffee, which costs $\$12$/lb, with Brazilian coffee, which costs $\$8$/lb. If they obtained $20$ pounds of mixture and its final…
In a coffee shop, they mixed Colombian coffee, which costs $\$12$/lb, with Brazilian coffee, which costs $\$8$/lb. If they obtained $20$ pounds of mixture and its final price is $\$9$/lb, how many pounds of each type of coffee did they use?
A perfume laboratory wants to create a new fragrance by mixing two essences. Essence A costs $\$120$/liter and essence B costs $\$75$/liter. They have decided that to…
A perfume laboratory wants to create a new fragrance by mixing two essences. Essence A costs $\$120$/liter and essence B costs $\$75$/liter. They have decided that to obtain a $250$ mL bottle, the cost of the mixture must be exactly $\$24,75$. What quantity of each essence, measured in milliliters, must the bottle contain?
Distance, rate, and time are linked by $d = r \cdot t$. In a catch-up problem, one traveler starts earlier. When the second traveler catches up, both are at the same spot, so both have gone the same distance. Let $t$ be the time of the traveler who started first. The follower has been moving for $t$ minus the head start.
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In a robotics competition, a robot leaves the starting line at a constant speed of $2$ ft/s. Thirty seconds later, a second robot sets off in pursuit at a speed of $3,5$…
In a robotics competition, a robot leaves the starting line at a constant speed of $2$ ft/s. Thirty seconds later, a second robot sets off in pursuit at a speed of $3,5$ ft/s. If the track has a total length of $150$ feet before reaching an obstacle zone, will the second robot manage to catch the first one before it enters the obstacle zone? Set up the equation and justify the answer by calculating the catch-up distance.
When $x^2$ survives: right triangles and work problems
Sometimes the $x^2$ terms do not cancel, and you get a quadratic equation. Two common cases are the Pythagorean Theorem and "working together" problems. First, two tools.
Expanding a square. $(x + a)^2 = x^2 + 2ax + a^2$. The middle term $2ax$ is easy to forget. Draw an $(x + a)$ by $(x + a)$ square if you need to see it.
Solving $x^2 + bx + c = 0$ by factoring. Find two numbers whose product is $c$ and whose sum is $b$. For $x^2 - x - 6$, the numbers $-3$ and $2$ have product $-6$ and sum $-1$, so $x^2 - x - 6 = (x - 3)(x + 2)$. (The quadratic formula also works. The Quadratic equations page covers it.)
Always check each solution against the story. A length or a time cannot be negative, so reject any solution that does not make sense.
Work problems. If a pipe fills a pool in $x$ hours, it fills $\tfrac{1}{x}$ of the pool each hour. Rates add, so if two pipes work together, their fractions per hour add up to the fraction per hour of the team.
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Calculate the perimeter of a right triangle knowing that its three sides are expressed by three consecutive natural numbers. (Hint: use the Pythagorean theorem…
Calculate the perimeter of a right triangle knowing that its three sides are expressed by three consecutive natural numbers. (Hint: use the Pythagorean theorem $a^2 + b^2 = c^2$, where $c$ is the longest side).
A water tank is filled by two faucets. Faucet A takes $6$ hours longer than faucet B to fill the tank separately. If both are opened at the same time, the tank fills in…
A water tank is filled by two faucets. Faucet A takes $6$ hours longer than faucet B to fill the tank separately. If both are opened at the same time, the tank fills in exactly $4$ hours. Work out how long it would take faucet B to fill the tank on its own. (This problem requires solving a quadratic equation).
Back to the start: Let $g$ = the number of extra gigabytes Priya used. The bill is $20 + 5g = 50$, so $5g = 30$ and $g = 6$. She used 6 extra gigabytes. Check: $20 + 5 \cdot 6 = 50$ ✓.