Exercises: Algebraic language

In the world of video games, we sometimes use formulas to calculate scores. Translate the following phrases into algebraic language, letting $x$ be the number of gems a player has obtained:

• a) Double the number of gems.

• b) Triple the gems plus $50$ extra points.

• c) The number of gems they would have if they lost $5$ gems.

• d) Half the gems they have.

Copy the following table into your notebook and complete the information about these monomials (remember that if there is no number in front of the letter, the coefficient is $1$):

Monomial	Coefficient	Literal part	Degree
$7a^2$
$-5x$
$x^3y$
$\frac{1}{2}b$

We want to fence off an area for a music festival that is square in shape. We do not know the length of the side yet, so we will call it $x$.

• a) Write an algebraic expression for the perimeter of the area (the sum of all its sides).

• b) Write an algebraic expression for the area of the area.

Calculate the numerical value of the following expressions for the given value (remember that between a number and a letter there is always an invisible multiplication):

• a) $3x + 5$ when $x = 2$

• b) $10 - 2n$ when $n = 3$

• c) $a^2 + 1$ when $a = 5$

• d) $\frac{x}{2} + 7$ when $x = 10$

Look at the following sequence of figures formed by orange dots. If we let $n$ be the position of the figure (Figure 1, Figure 2...):

• a) How many dots will Figure 4 have?

• b) Write an algebraic expression that tells us how many dots are in Figure $n$.

In a clothes shop, a hoodie costs $s$ euros. If a pair of trousers costs $12$ euros less than the hoodie, express in algebraic language:

• a) The price of the trousers.

• b) The total price of buying two hoodies.

• c) The total price of buying one hoodie and a pair of trousers.

Let $x$ be Ioana's current age. Express her family's ages algebraically:

• a) The age Ioana will be in $8$ years' time.

• b) The age Ioana was $3$ years ago.

• c) Her brother is twice Ioana's age. How old is her brother?

• d) Her mother is three times the age Ioana will be in $1$ year's time.

If $n$ represents any integer, write the algebraic expression for:

• a) The number after $n$.

• b) The number before $n$.

• c) Triple the number after $n$.

• d) The sum of a number and the number before it.

Cian has to translate into algebraic language the following instruction given by his teacher: "Three times the sum of a number plus four". In his notebook he has written the expression:

$$3 \times x + 4$$

1. a) Identify the mistake made by Cian and explain with reasoning what everyday language phrase actually corresponds to what he wrote.
2. b) Write the correct algebraic translation using the necessary mathematical punctuation marks.
3. c) Check the error numerically by substituting $x = 2$ into both Cian's incorrect expression and your correct one. Why do they give different values and what does this show about the need to use parentheses?

In a Mathematics class, the following inverse problem is proposed: from the already constructed algebraic expression $2 \times (x - 5)$, you must reconstruct the original problem.

1. a) Translate this expression into an ordinary language sentence, rigorously using the words "double", "difference", and "number".
2. b) Model a plausible everyday situation (for example, about discounted shopping or points scored in a game) that exactly matches this expression, clearly stating what the variable $x$ represents.
3. c) If we know that the final numerical value of the expression after carrying out the calculations is $16$, deduce what the initial value of $x$ was by working backwards through the operations (in reverse order).

In a popular adventure video game, players earn different scores depending on their actions. If we let $x$ be the number of gold chests found and $y$ be the number of side quests completed, express the following situations in algebraic language:

• Triple the number of gold chests found.

• The number of side quests plus five extra bonus quests.

• Double the sum of the gold chests and side quests.

• Half of the chests found after losing three of them in a battle.

• The total points if each chest is worth 100 points and each quest is worth 50 points.

Calculate the perimeter of the following padel court whose dimensions are given in metres using algebraic expressions. Once you have obtained the simplified expression, calculate its numerical value for $x = 4$ metres.

A TikTok content creator is analysing their monthly income. If $v$ represents the number of videos published in a month, their earnings in euros are defined by the algebraic expression: $G = 0.5 \times v + 12$.

• What do the numbers $0.5$ and $12$ represent in this context?

• What will their earnings be if this month they managed to publish a total of $v = 40$ videos?

• If next month they decide to publish twice as many videos as this month, write the new expression for their earnings.

Look at the following sequence of hexagonal tables placed in a line for a school celebration. Each hexagon represents a table where students sit.

If one table ($n=1$) seats 6 people, two tables ($n=2$) seat 10 people and three tables ($n=3$) seat 14 people:

• Create an algebraic expression to calculate the number of people who can sit as a function of the number of tables $n$.

• Calculate how many people can sit if 15 tables are joined together.

In a sports shop, a football shirt costs $x$ euros and a pair of shorts costs $y$ euros. Translate the following expressions from algebraic language into plain English:

• $x + y$

• $2x + 3y$

• $x - 10$

• $0.85 \times x$

• $3(x + y)$

Copy and complete the following table about monomials in your notebook. Remember that if no coefficient is shown, it is understood to be 1.

Monomial	Coefficient	Literal Part	Degree
$-5x^2$
$ab^3c$
$\frac{3}{4}x^4y$
$-z$
$12$

Amina, Borja and Clara are comparing how many followers they have on social media. Borja has twice as many followers as Amina. Clara has 500 fewer followers than Borja. If we let $s$ be the number of followers Amina has:

• Express Borja's followers in algebraic language.

• Express Clara's followers in algebraic language.

• Write an expression representing the total number of followers of the three friends.

• If Amina has $s = 1200$ followers, how many followers do all three have altogether?

Valentina suggests a magic trick to her brother using algebraic language. Follow the steps and then explain algebraically why the same result is always obtained:

1. Think of any number ($n$).

2. Add 5 to it.

3. Multiply the result by 2.

4. Subtract 10 from the value obtained.

5. Divide the result by 2.

6. What number do you get compared to the initial number? Write the simplified expression for the process.

In the public taxi service of an Irish city, official daytime fares are calculated by combining several components:

• A base fare (fixed cost for starting the ride) of $2.50$ €.

• A cost of $1.30$ € for each kilometer travelled.

• An additional fixed supplement of $3.20$ € if the ride ends at the central train station.

1. a) Write the algebraic expression that allows calculating the total cost of an ordinary ride of $k$ kilometers (without the station supplement).
2. b) Write the algebraic expression for a ride of $k$ kilometers that ends at the train station, and simplify it by adding the two fixed amounts into a single term.
3. c) Two coworkers share the taxi to the station and agree to split the total amount exactly equally. Write the algebraic expression for the price each must pay in terms of $k$, and calculate how much each will pay if the total ride is $10$ km.

To renew the railing of a pedestrian footbridge in the Tablas de Daimiel National Park, workers place vertical posts connected by wooden crossbeams forming continuous modules:

• For $1$ meter of footbridge, $6$ pieces of wood are needed.

• For $2$ meters of footbridge, $10$ pieces of wood are needed.

• For $3$ meters of footbridge, $14$ pieces of wood are needed.

• For $4$ meters of footbridge, $18$ pieces of wood are needed.

1. a) Estimate and check: before deducing any formula, if the workers must build a $25$-meter stretch, make a quick mental estimate: do you think they will need more or less than $100$ pieces of wood? Justify your estimate without doing the step-by-step calculation.
2. b) Observe the pattern: how many pieces of wood are added for each additional meter? Deduce with reasoning the algebraic expression that gives the total number of pieces of wood needed for a stretch of $n$ meters of footbridge.
3. c) Use your expression to calculate the exact number of pieces of wood that will be used for the $25$ meters. Compare the result obtained with your estimate from part a) and evaluate whether it was accurate and reasonable.

In a garden design company, hexagonal tiles are used to surround flowerbeds. If we place the flowerbeds in a line, each additional flowerbed shares some tiles with the previous one to save material. Look at the following design sequence:

If we call $n$ the number of flowerbeds we want to place in a row:

1. a) Find an algebraic expression to calculate the number of tiles needed for $n$ flowerbeds.
2. b) Use this expression to calculate how many tiles would be needed for a row of $25$ flowerbeds.

Express the following complex statements in algebraic language, using a single variable $x$ for the unknown number:

1. a) Three times the difference between the square of a number and its half.
2. b) The sum of three consecutive natural numbers, knowing that the first one is $x-2$.
3. c) The final price of a hoodie that cost $x$ euros, if a $10\%$ increase for delivery costs is applied first and then a discount of $5$ euros from a voucher.
4. d) The difference between the cube of the next consecutive number and twice its previous number.

A group of friends wants to buy a video games console as a gift for a classmate. The console costs $C$ euros. They decide that two of the friends, who are siblings, will each pay only half of what each of the others pays.

1. a) If $x$ is the amount of money contributed by each of the non-sibling friends, and there are a total of $6$ friends (including the two siblings), express the total cost $C$ of the console algebraically.
2. b) If the cost of the console is $300$ euros, calculate how much money each of the siblings must contribute.

In the design of a modern swimming pool, a rectangular water area of length $l$ and width $a$ is to be surrounded by a wooden border $2$ metres wide all around its perimeter.

1. a) Write the algebraic expression representing the total area occupied by the wood (the border).
2. b) Simplify the expression obtained in the previous section.
3. c) What would be the numerical value of that area if the pool is $10$ metres long and $5$ metres wide?

Consider the algebraic expression $P(x) = \frac{x^2 - 4}{x + 2}$.

1. a) Calculate the numerical value of $P(x)$ for $x = 0, x = 2$ and $x = 10$.
2. b) Looking at the results, could you suggest a much simpler expression that is equivalent to $P(x)$ for any value of $x$ (except for $x = -2$)?
3. c) Why do you think it is not possible to calculate the numerical value of $P(x)$ for $x = -2$? Justify your answer based on what you know about fractions.

In a triathlon competition, an athlete runs at a constant speed of $v$ km/h.

1. a) Express algebraically the distance they cover in a time of $t$ hours and $15$ minutes. (Remember to express the time in a single unit).
2. b) If the athlete decides to increase their speed by $2$ km/h during the second half of the time, write the expression for the total distance covered in a total time of $t$ hours.
3. c) Calculate the total distance from part b) if $v = 12$ km/h and $t = 2$ hours.

A magician asks a spectator to follow these instructions:

1. Think of any natural number $n$.

2. Multiply it by $6$.

3. Add $12$ to the previous result.

4. Divide everything by $3$.

5. Subtract twice the number you originally thought of.

1. a) Translate each step of the trick into algebraic language and write the final simplified expression.
2. b) What is the final result of the trick? Does it depend on the number the spectator thought of? Justify your answer using algebraic language.

In a logistics warehouse, boxes are stacked to form pyramid structures with a square base. At the top level (level 1) there is $1$ box. At level 2 (below the first) there are $4$ boxes forming a $2 \times 2$ square. At level 3 there are $9$ boxes forming a $3 \times 3$ square.

1. a) Write the algebraic expression that represents the number of boxes at level $n$.
2. b) Write an algebraic expression that represents the total number of boxes if the pyramid has only $3$ levels.
3. c) If a pyramid has $n$ levels, the total number of boxes is known to be given by the formula $T = \frac{n(n+1)(2n+1)}{6}$. Calculate how many boxes there will be in total in a $10$-level structure.

In a recreational mathematics workshop, the following number magic trick is proposed:

1. Think of any integer.

2. Add $5$ to it.

3. Multiply the result obtained by $2$.

4. Subtract $6$ from it.

5. Divide the result by $2$.

6. Subtract the number you had thought of at the beginning.

The final result is always $2$, no matter the chosen number!

To convince the class that this is not a coincidence, two approaches are discussed:

• Method A (Empirical verification): Test the trick with five very different numbers (a single-digit number, a three-digit number, zero, a negative number, and a fraction).

• Method B (Algebraic proof): Call the initial number $n$, algebraically chain all the operations into a single expression, and simplify it step by step.

1. a) Fully develop Method B: write the resulting algebraic expression after each of the 6 instructions and simplify it to prove that the final result is $2$.
2. b) Choose the method and justify: compare both approaches, explaining why Method A, despite testing many cases, does not prove that the trick always works, and what definitive advantage the algebraic language of Method B provides.

During a review activity, the teacher asks students to translate the following sentence into algebraic language: "Three times the predecessor of a number $x$, increased by twice the successor of that number $x$".

Two students hand in different proposals:

• Cian writes: $3 \times x - 1 + 2 \times x + 1$

• Aoife writes: $3 \times (x - 1) + 2 \times (x + 1)$

1. a) Analyze Cian's proposal: identify the two mistakes made by not using parentheses and explain what everyday sentence would correspond word-for-word to the expression he wrote.
2. b) Expand and simplify Aoife's expression by applying the distributive property and grouping like terms to obtain a reduced expression of the form $a \times x + b$.
3. c) Reason and justify: could it be possible that, with $x$ being an integer, the final value of Aoife's expression is equal to $40$? Set the reduced expression equal to $40$, solve for $x$, and justify your conclusion by analyzing whether the result belongs to the set of integers.