Exercises: Word problems with equations

In the Science lab we have a balance in equilibrium, as the figure shows. On the left pan there are $2$ identical boxes (each weighing $x$ kg) and a $1$ kg weight. On the right pan there is a $7$ kg weight. Write the equation that represents this equilibrium and calculate how much each box weighs.

Cian is playing a video game and has scored twice as many points in the current level as in the previous level. If he scores a total of $150$ points across both levels, how many points did he get in each level? Let $x$ be the points in the first level.

We are designing an esports team logo shaped like an isosceles triangle. We want the perimetre to be $30$ cm. If the base measures $10$ cm and the other two sides are equal, how long must each of those sides ($x$) be?

If you add a consecutive number (the number that comes right after) to a number, the result is $43$. Set up an equation to find out what the number is.

Amina is $3$ years older than her younger brother. If their combined age is $19$ years, how old is each of them? Let $x$ be the age of the younger brother.

A group of friends buys a family-sized pizza that costs $12$ €. If they have $8$ € left over after paying for it from the money they had collected, how much money did they have at the start? Represent the initial amount with the letter $x$.

We want to fence off a small rectangular vegetable patch to plant tomatoes. The length of the patch is twice its width. If we have used $18$ metres of fencing to completely surround it (perimetre), what are the width and length of the patch?

On a shelf in a comic book shop, there are three times as many action manga as mystery manga. If there are $48$ comics in total between both genres, how many are there of each type? Let $x$ be the mystery comics.

In a 1st Year exam, Cian has to solve this problem: “The sum of three consecutive integers is $48$. What are these three numbers?”. Cian writes the following on his sheet:

“I call the first number $x$. Since they are consecutive, the three numbers are $x$, $2x$, and $3x$.

Equation: $x + 2x + 3x = 48 \implies 6x = 48 \implies x = 8$.

Answer: The numbers are $8$, $16$, and $24$”.

1. a) Locate and explain with reasoning the conceptual error Cian made when translating the expression “three consecutive integers”.
2. b) Write the correct equation, precisely defining what the unknown $x$ represents.
3. c) Solve the equation, state what the three correct numbers are, and check that their sum equals $48$.

At the neighborhood stationery shop, you buy a binder and $3$ identical notebooks for the school year. The binder costs exactly twice as much as a notebook, and you pay a total of $12.50$ € at the checkout.

1. a) Before setting up any equation, reason whether it is possible that each notebook costs $3$ €. Make a quick estimate of between which two integer values in euros the price of a notebook must lie.
2. b) Define the unknown $x$ indicating its units, set up a linear equation, and calculate the exact price of a notebook and of the binder.
3. c) Check the solution obtained in the problem statement and explain whether your estimate in part a) was accurate.

Three friends are comparing how many songs they have in their favourite playlist. Aoife has twice as many songs as Seán, and Niamh has 15 more songs than Seán. If all three have a total of 155 songs altogether, how many songs does each person have in their playlist? Set up an equation where $x$ is the number of Seán's songs.

At Cian's birthday party, two types of invitations were handed out: digital and paper. The number of digital invitations was three times the number of paper ones. If a total of 48 invitations were sent out, how many people received a paper invitation and how many a digital one?

Aoife bought a video game on offer and a new controller. The controller cost her $25$ € less than the video game. If she paid a total of $89$ € for both items, what is the original price of each item without counting other discounts?

In a shopping centre car park there is a total of $32$ vehicles, including cars and motorbikes. If we count all their wheels (excluding spare tyres), we get a total of $94$ wheels. Calculate how many cars and how many motorbikes are in the car park.

(Hint: If we let $x$ be the number of cars, the number of motorbikes will be $32 - x$).

The PE teacher has bought three boxes of tennis balls. The second box contains twice as many balls as the first, and the third box has $10$ more balls than the second. If there are $85$ balls in total, how many balls are in each box?

Seán is 5 years older than his sister Aoife. In 8 years' time, the sum of their ages will be 41 years. How old is each of them now?

(Remember to add 8 years to each person's age to set up the future sum).

To balance a scale in the Science lab, we have placed a stone of unknown weight $x$ and a $12$-gram weight on the left pan. On the right pan, to keep it balanced, we had to put four equal weights, each with the same weight $x$ as the original stone. How much does the stone weigh?

The transport authority of an Irish city (Cork) offers two fares for young students:

• Youth Fare: a fixed monthly fee of $10$ € is paid and each trip taken costs $0.40$ €.

• Single Ticket: no fixed monthly fee is paid, but each trip costs $0.90$ €.

1. a) Model the situation mathematically: define the unknown $x$ with its meaning and set up an equation expressing for what number of monthly trips the total cost of both fares is exactly the same.
2. b) Solve the equation set up and state the exact number of trips at the break-even point.
3. c) Interpret the result: if a student knows that she will make $25$ trips a month to go to secondary school and her sports activities, which of the two fares is cheaper for her? Justify your answer by calculating the expense she would have with each option.

Imagine that you are part of the teaching team and you have been asked to design a geometry problem for the upcoming 1st Year exam. The problem must be about the perimeter of a rectangular plot and must strictly meet these three conditions:

1. The base of the plot must measure $4\text{ m}$ more than three times its height.

2. The unknown $x$ must represent the height of the plot in meters.

3. When solving the perimeter equation, the solution must necessarily be $x = 6\text{ m}$.

1. a) Write the simplified algebraic expression for the base in terms of $x$ and the simplified algebraic expression for the total perimeter of the rectangle.
2. b) Use the value $x = 6\text{ m}$ to deduce what number must appear as the total perimeter in the text of the problem.
3. c) Write the final statement of the problem just as your classmates would read it on the exam, clearly formulating the data and the final question.

On a scientific expedition through the Amazon, the team consumes $\frac{1}{4}$ of their water supply on the first day. On the second day, they consume $\frac{2}{5}$ of what was left in the tank. If after these two days they still have $135$ litres, how many litres of water did they have at the start of the expedition?

The perimeter of an isosceles triangle-shaped technology park is $110$ metres. We know that each of the equal sides measures $10$ metres more than twice the base. Calculate the length of each of the three sides of the park by setting up an equation.

In a games shop, they sold three times as many copies of an adventure game as a sports game. Furthermore, they sold $15$ fewer units of a platform game than the adventure game. If they sold a total of $125$ video games across the three titles, how many copies were sold of each one?

A digital content creator reviewed their monthly statistics. The comedy video has twice as many views as the technology video, and the travel video has $4000$ more views than the comedy video. If all three videos total $24000$ views altogether, how many views does each video have?

Salma's mother is currently three times her age. In $12$ years, her mother's age will be only twice Salma's age. How old is each of them currently? Use a table to organise the data for the present and the future.

In a piggy bank, there are $0.50$ € and $1$ € coins. In total, there are $35$ coins with a total value of $26$ €. How many coins of each type are in the piggy bank? (Hint: if you let $x$ be the number of $1$ € coins, the number of $0.50$ € coins will be $35 - x$).

Three siblings, Seán, Aoife and Cian, want to buy a present for their parents that costs $105$ €. Seán contributes an amount of money, Aoife gives twice as much as Seán, and Cian gives $15$ € less than Aoife. How much money does each person contribute so that they have exactly enough for the gift?

An olive oil cooperative in Kilkenny distributes $45\,000$ € in annual profits among its three founding partners. Due to their level of dedication, the second partner receives twice as much as the first, and the third partner receives exactly half the sum of what the other two earn together.

To calculate how much money corresponds to each one, two students propose different approaches:

• Aoife's approach: she calls $x$ the amount the first partner receives, expresses what the other two receive in terms of it, and sets up the equation by adding the three parts.

• Niamh's approach: she calls $y$ the amount the third partner receives and deduces that the sum of the first and second partners is equal to $2y$.

1. a) Set up and solve the equation following Aoife's method, indicating step by step the amount that corresponds to each partner.
2. b) Set up and solve the equation following Niamh's method. Check whether the same final amounts are obtained for each person.
3. c) Compare both methods and reflect: what mathematical advantage does Niamh's approach offer when solving the first equation? Which of the two methods do you consider more intuitive and direct to avoid getting stuck if you do not spot the hidden relationship between the parts? Justify your answer.

In a family home in Galway, the monthly electricity bill consists of a fixed charge of $14$ € (for contracted power) and a variable charge of $0.18$ € for each $\text{kWh}$ consumed. In the month of January, the total amount of the bill before taxes came to $64.40$ €.

1. a) Before solving using algebra, make a reasoned mental estimate: will the consumption have been closer to $200\text{ kWh}$, $300\text{ kWh}$, or $400\text{ kWh}$? Justify how you deduce it by calculating approximate values mentally.
2. b) Define the unknown $x$ with its units, set up a linear equation, and calculate the exact consumption in $\text{kWh}$ for the month of January.
3. c) In spring, the family adopts energy-saving habits and manages to reduce their consumption to exactly half the $\text{kWh}$ of January, while maintaining the same contracted power. A family member claims: “Since we consumed half the electricity, this month's bill will be exactly half ($32.20$ €)”. Set up the mathematical expression for the new cost and explain algebraically why that statement is false, clarifying what role the fixed charge plays in the proportionality of the bill.

Unit challenge. Which tariff suits us best at home? Investigate and optimize your family's electricity bill by applying linear equations.

1. Data collection: Check a real electricity bill from your home or visit the Commission for Regulation of Utilities (CRU) or Bonkers.ie website. Record the monthly fixed cost and the price per kWh (for example, $0.14$ €/kWh). Then, choose an alternative tariff from another company with a different fixed and variable cost.

2. Algebraic modeling: Set up a word problem and its linear equation to find out at what monthly consumption in kWh ($x$) both options would cost exactly the same: $F_1 + V_1 \times x = F_2 + V_2 \times x$. Solve the equation, detailing and justifying each algebraic step.

3. Simulation and proposal: Design a table in a spreadsheet comparing both tariffs for different consumptions ($100$, $200$, and $300$ kWh) and check your solution. Write a brief report giving a reasoned recommendation for the best option based on your household's actual consumption.

Assessment criteria:

• Appropriate extraction and organization of real consumption and tariff data.

• Correct formulation and step-by-step resolution of the linear equation.

• Use of the spreadsheet to verify solutions and simulate scenarios.

• Coherent and critical justification of the final savings proposal.