You and a friend start biking toward each other from opposite ends of a 36-mile trail. You ride 10 mph and your friend rides 8 mph. When do you meet? You could guess and check, but one short equation does it faster. Solving an equation is mostly mechanical. The hard part, and the part that really counts, is setting it up.
The Four-Step Method
Define the unknown. Write down what the variable means: "$x$ = Priya's age today." If there are several unknown quantities, try to write them all using the same variable.
Translate each fact into an algebraic expression.
Set up an equation using the condition the story gives you, then solve it.
Check your answer against the story (not against your equation, which might be wrong), and answer in a full sentence with units.
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In an eSports tournament, Alex's team has scored three times as many points as Lucy's team. If between the two teams they have scored a total of $152$ points, how many…
In an eSports tournament, Alex's team has scored three times as many points as Lucy's team. If between the two teams they have scored a total of $152$ points, how many points has each team obtained?
A food delivery app rider is carrying twice as many pizzas as burgers in their backpack. If they are transporting a total of $12$ orders between pizzas and burgers, how…
A food delivery app rider is carrying twice as many pizzas as burgers in their backpack. If they are transporting a total of $12$ orders between pizzas and burgers, how many of each kind are there?
Lucy has started saving to buy a tablet that costs $\$240$. She has already saved $\$60$ and, from now on, she is going to put away $\$12…
Lucy has started saving to buy a tablet that costs $\$240$. She has already saved $\$60$ and, from now on, she is going to put away $\$12$ of her allowance every week. How many weeks will it take her to have enough money for the purchase?
Many problems are about unknown numbers. Learn to translate the key phrases:
Phrase
Expression
a number plus 5 / 5 more than a number
$x + 5$
3 less than a number
$x - 3$
twice a number
$2x$
the square of a number
$x^2$
three consecutive integers
$x,\ x + 1,\ x + 2$
consecutive even (or odd) integers
$x,\ x + 2,\ x + 4$
The same idea works for prices and amounts. If a standing ticket costs $\$15$ more than a seat ticket, call the seat price $x$. The standing price is $x + 15$.
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A group of friends buys tickets for a concert. The "standing" tickets cost $\$15$ more than the "seated" ones. If they buy one of each type and the total bill is $\$85$,…
A group of friends buys tickets for a concert. The "standing" tickets cost $\$15$ more than the "seated" ones. If they buy one of each type and the total bill is $\$85$, what is the price of each ticket?
The sum of three race bib numbers, which happen to be three consecutive integers, is equal to $63$. What numbers are these three runners wearing on their shirts?
The sum of three race bib numbers, which happen to be three consecutive integers, is equal to $63$. What numbers are these three runners wearing on their shirts?
In the number 74, the 7 is in the tens place and the 4 is in the ones place. So $74 = 10 \cdot 7 + 4$. In general, a two-digit number with tens digit $x$ and ones digit $y$ is worth $10x + y$. Reverse the digits and you get $10y + x$.
There are two unknowns, $x$ and $y$. The problem always gives you a relation between them, like "the digits add up to 11." Use it to write $y$ in terms of $x$, then substitute.
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Find a two-digit number knowing that the sum of its digits is $12$ and that, if the order of its digits is reversed, the new number obtained exceeds the original by $18$…
Find a two-digit number knowing that the sum of its digits is $12$ and that, if the order of its digits is reversed, the new number obtained exceeds the original by $18$ units. Represent the original number as $10x + y$.
Make a table with columns for "now," "in … years," or "… years ago." Time passes the same for everyone, so you add (or subtract) the same number from every person's age.
Now
In 4 years
Priya
$x$
$x + 4$
Marcus (5 years older)
$x + 5$
$x + 9$
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Emma is $4$ years older than her little brother, Ethan. In $6$ years' time, the sum of their ages will be $32$ years. Calculate the current age of each one by setting up…
Emma is $4$ years older than her little brother, Ethan. In $6$ years' time, the sum of their ages will be $32$ years. Calculate the current age of each one by setting up a linear equation.
Currently, a father's age is three times that of his daughter. In $12$ years' time, the father's age will only be double that of his daughter. How old is each of them…
Currently, a father's age is three times that of his daughter. In $12$ years' time, the father's age will only be double that of his daughter. How old is each of them currently?
Sometimes an $x^2$ term survives when you simplify. Move everything to one side to get a quadratic equation $ax^2 + bx + c = 0$. Then use one of two tools.
Tool 1: factor. If $(x - p)(x - q) = 0$, then $x - p = 0$ or $x - q = 0$. A product can only be zero if one of its factors is zero, so $x = p$ or $x = q$.
The solutions of a quadratic equation are the $x$-intercepts of its graph. Here is $y = x^2 - 3x - 10$, which crosses the $x$-axis twice.
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Find a natural number such that the difference between its square and three times the number is equal to $40$. Is the solution to this problem unique in the set of…
Find a natural number such that the difference between its square and three times the number is equal to $40$. Is the solution to this problem unique in the set of natural numbers? Justify your answer by setting up the corresponding equation.
A mother's current age is the square of her daughter's age. In $9$ years' time, the mother's age will only be three times the age her daughter is at that time. Find the…
A mother's current age is the square of her daughter's age. In $9$ years' time, the mother's age will only be three times the age her daughter is at that time. Find the current ages of both.
Always draw the figure and label its sides using $x$. Then use the right formula: perimeter, area, the Pythagorean Theorem, and so on.
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A rectangular school vegetable garden is to be fenced to protect the crops. We know that the length of the garden measures $5$ feet more than the width and that $38$…
A rectangular school vegetable garden is to be fenced to protect the crops. We know that the length of the garden measures $5$ feet more than the width and that $38$ feet of fencing were needed to completely surround it. What are the dimensions of the garden?
Calculate the side length of a square advertising panel knowing that if its side were to increase by $2$ feet, its area would increase by $16$ square feet.
Calculate the side length of a square advertising panel knowing that if its side were to increase by $2$ feet, its area would increase by $16$ square feet.
A new rectangular play area is being designed in a municipal park. It is known that the length of the area must be $5$ feet greater than the width. If the total area…
A new rectangular play area is being designed in a municipal park. It is known that the length of the area must be $5$ feet greater than the width. If the total area intended for the games is $150$ $m^2$, calculate the dimensions (length and width) that this area will have.
When a border of width $x$ surrounds a rectangle, it adds $x$ on both sides. Each dimension grows by $2x$, not $x$.
Right triangles and isosceles triangles
For a right triangle with legs $a$ and $b$: area $= \tfrac{1}{2}ab$ and $a^2 + b^2 = c^2$. For an isosceles triangle, the height splits the base in half. That gives a right triangle: $(\text{half the base})^2 + h^2 = (\text{equal side})^2$.
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Calculate the dimensions of an isosceles triangle knowing that the unequal side measures $4$ $cm$ less than each of the equal sides and that the total perimeter is $26$…
Calculate the dimensions of an isosceles triangle knowing that the unequal side measures $4$ $cm$ less than each of the equal sides and that the total perimeter is $26$ $cm$. Once the sides have been found, calculate the area of the triangle.
A graphic designer is working on the logo for a new company. The design consists of a central rectangle for the name, surrounded by a decorative border of uniform…
A graphic designer is working on the logo for a new company. The design consists of a central rectangle for the name, surrounded by a decorative border of uniform thickness $x$. The inner rectangle has dimensions of $12\text{ cm}$ by $8\text{ cm}$. If the total area of the logo (including the border) must be exactly double the area of the inner rectangle, what must the thickness $x$ of this border be?
The area of a right triangle is $30\text{ cm}^2$ and its hypotenuse measures $13\text{ cm}$. Calculate the length of the two shorter sides of the triangle.
The area of a right triangle is $30\text{ cm}^2$ and its hypotenuse measures $13\text{ cm}$. Calculate the length of the two shorter sides of the triangle.
Everything here comes from one formula: distance = rate × time, or $d = rt$. Use these setups:
Situation
Equation
Meeting (moving toward each other, leaving together)
$r_1 t + r_2 t = \text{total distance}$
Catching up (same direction)
$r_1 t = \text{head start} + r_2 t$
Delayed start (one leaves later)
the later one travels for $t - \text{delay}$
Why does meeting work? When they meet, the two distances traveled together cover the whole gap between the starting points.
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Two cyclists, Lucy and Marcus, are at a distance of $60$ $km$ and decide to meet at an intermediate point. Lucy sets off from $A$ towards Marcus at a constant speed of…
Two cyclists, Lucy and Marcus, are at a distance of $60$ $km$ and decide to meet at an intermediate point. Lucy sets off from $A$ towards Marcus at a constant speed of $20$ $km/h$. At the same time, Marcus sets off from $B$ towards Lucy at a speed of $10$ $km/h$. How long will it take them to cross paths and at what distance from point $A$ will they meet?
Two cities, $A$ and $B$, are separated by a distance of $155\text{ km}$. At 10:00, a cyclist leaves $A$ towards $E$ at a constant speed of $20\text{ km/h}$. An hour and…
Two cities, $A$ and $B$, are separated by a distance of $155\text{ km}$. At 10:00, a cyclist leaves $A$ towards $E$ at a constant speed of $20\text{ km/h}$. An hour and a half later, a motorcycle leaves $E$ towards $A$ at a speed of $80\text{ km/h}$. At what time will they cross paths and at what distance from city $A$ will they meet?
All these problems share one pattern. Take pieces of two things, and the total value is the sum of the values of the pieces.
Type
Equation
Price mixture
$p_1 x + p_2 (T - x) = p_{\text{mix}} \cdot T$
Percent solution
$r_1 x + r_2 (T - x) = r_{\text{mix}} \cdot T$
Two investments
$r_1 x + r_2 (T - x) = \text{total interest}$
Here $x$ is the amount of the first thing and $T - x$ is the amount of the second, where $T$ is the total. Convert percents to decimals before you calculate.
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In a gourmet shop, they want to mix two types of coffee. "Arabica" coffee is priced at $\$12$/lb and "Robusta" coffee costs $\$8$/lb. If we want to obtain a mixture of…
In a gourmet shop, they want to mix two types of coffee. "Arabica" coffee is priced at $\$12$/lb and "Robusta" coffee costs $\$8$/lb. If we want to obtain a mixture of $20$ $kg$ that works out at a price of $\$10,50$/lb, how many pounds of each type of coffee must we use?
In a chemical laboratory, $20\text{ liters}$ of an acid solution at $25\%$ are needed. The technician has two jugs available: one with a mixture at $10\%$ acid and…
In a chemical laboratory, $20\text{ liters}$ of an acid solution at $25\%$ are needed. The technician has two jugs available: one with a mixture at $10\%$ acid and another with a concentrated mixture at $50\%$. How many liters from each jug must be mixed to obtain the desired concentration?
An investor has $12\,000\text{ \$}$ which they divide into two parts to deposit into two different financial products. The first offers an annual return of $4\%…
An investor has $12\,000\text{ \$}$ which they divide into two parts to deposit into two different financial products. The first offers an annual return of $4\%$, while the second, which is riskier, offers an annual return of $7\%$. If at the end of the year the total profits obtained were $630\text{ \$}$, what amount did they allocate to each product?
If a cost $C$ is split among $x$ people, each pays $\dfrac{C}{x}$. When more people join, the share per person drops. Write: (share before) − (share after) = the difference.
Working together
If one person finishes a job in $x$ hours, they do $\dfrac{1}{x}$ of the job per hour. That is their rate. When people work together, their rates add. If they finish in $T$ hours together, the combined rate is $\dfrac{1}{T}$.
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A group of friends wants to buy a birthday present that costs $\$120$. At first there were a certain number of people, but at the last minute $2…
A group of friends wants to buy a birthday present that costs $\$120$. At first there were a certain number of people, but at the last minute $2$ more friends joined in, which meant that everyone had to pay $\$5$ less. How many friends were there initially in the group?
Two pipes connected to a tank can fill it together in $4\text{ hours}$. If used separately, the pipe with the smaller diameter takes $6\text{ hours}$ longer than the…
Two pipes connected to a tank can fill it together in $4\text{ hours}$. If used separately, the pipe with the smaller diameter takes $6\text{ hours}$ longer than the pipe with the larger diameter to fill the entire tank. How long would each pipe take on its own?
Back to the start: Emma and Liam, 36 miles apart at 10 mph and 8 mph, meet after $10t + 8t = 36$, so $t = 2$ hours. Setting up the equation took one line.
Let's practice! 8 medium exercises in this section