Exercises: Algebraic language

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

• a) Twice the number of gems.

• b) Three times the gems plus $50$ extra points.

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

• d) Half of 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 an enclosure for a music festival that has the shape of a square. We do not know how long the side is yet, so we will call it $x$.

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

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

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 $x = 10$

Observe the following series of figures formed by orange dots. If we call $n$ 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 there are in Figure $n$.

In a clothing store, a hoodie costs $s$ dollars. If a pair of pants costs $12$ dollars less than the hoodie, express in algebraic language:

• a) The price of the pants.

• b) The total price of buying two hoodies.

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

We call $x$ Mia's current age. Express algebraically the ages of her family:

• a) The age Mia will be in $8$ years.

• b) The age Mia was $3$ years ago.

• c) Her brother is twice Mia's age. What age is the brother?

• d) Her mother is three times the age Mia will be in $1$ years.

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

• a) The next number after $n$.

• b) The previous number before $n$.

• c) Three times the next number after $n$.

• d) The sum of a number and its previous number.

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

• Three times the gold chests found.

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

• Twice the sum of the gold chests and the 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 pickleball court whose dimensions are given in feet by algebraic expressions. Once the simplified expression is obtained, calculate its numerical value for $x = 4$ feet.

A TikTok content creator analyzes their monthly income. If $v$ represents the number of videos published in a month, their earnings in dollars are defined by the algebraic expression: $G = 0,5 \cdot v + 12$.

• What do the numbers $0,5$ and $12$ represent in this context?

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

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

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

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

• Create an algebraic expression that allows calculating the number of people who can sit depending on the number of tables $n$.

• Calculate how many people will be able to sit if 15 tables are joined together.

In a sporting goods store, a soccer jersey costs $x$ dollars and a pair of shorts costs $y$ dollars. Translate from algebraic language to ordinary language what the following expressions mean:

• $x + y$

• $2x + 3y$

• $x - 10$

• $0,85 \cdot x$

• $3(x + y)$

Copy and complete the following table about monomials in your notebook. Remember that if the coefficient does not appear, it is understood to be one.

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

Sophia, Brandon, and Chloe are comparing how many followers they have on social media. Brandon has twice as many followers as Sophia. Chloe has 500 fewer followers than Brandon. If we call $s$ the number of followers Sophia has:

• Express in algebraic language the followers Brandon has.

• Express in algebraic language the followers Chloe has.

• Write an expression that represents the sum of the followers of the three friends.

• If Sophia has $s = 1200$ followers, how many followers do they have among the three of them?

Emma proposes 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 of the process.

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. Observe the following design sequence:

Express the following complex sentences 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 of them is $x-2$.
3. c) The final price of a hoodie that cost $x$ dollars, if first a $10\%$ percent increase is applied for shipping costs and then a discount of $5$ dollars for a coupon.
4. d) The difference between the cube of the next number and twice its previous number.

A group of friends wants to give a video game console to a classmate. The console costs $C$ dollars. They decide that two of the friends, who are brothers, will only pay half of what each of the rest pays.

1. a) If $x$ is the amount of money contributed by each of the friends who are not brothers, and there is a total of $6$ friends (including the two brothers), express algebraically the total cost $C$ of the console.
2. b) If the cost of the console is $300$ dollars, calculate how much money each of the brothers 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 $2$ feet wide all around its perimeter.

1. a) Write the algebraic expression that represents 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 $10$ feet long and $5$ feet 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) Observing the results, could you propose 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 the numerical value of $P(x)$ cannot be calculated for $x = -2$? Reason your answer based on what you know about fractions.

In a triathlon competition, an athlete runs at a constant speed of $v$ mph.

1. a) Express algebraically the distance they travel 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 $2$ mph during the second half of the time, write the expression for the total distance travelled in a total time of $t$ hours.
3. c) Calculate the total distance from part b) if $v = 12$ mph 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 thought of at the beginning.

1. a) Translate each step of the trick into algebraic language and write the simplified final 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 forming square-based pyramidal structures. On the highest level (level 1) there is $1$ box. On level 2 (below the first) there are $4$ boxes forming a $2 \times 2$ square. On level 3 there are $9$ boxes forming a $3 \times 3$ square.