Problems with equations

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.

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 $x$.

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.

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.

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.

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.

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).

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?

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?

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.

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.

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?

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?

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.

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 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.