The Language of Algebra

Express the following situations related to savings and social media in algebraic language:

1. a) The number of followers an influencer has, $x$, increased by $150$.
2. b) The price of a movie ticket, $p$, reduced by $40$ pesos.
3. c) Three times my brother's age, if his age is $y$.
4. d) Half of the money I have in my piggy bank, where $d$ is the total amount.

Match each phrase with its corresponding algebraic expression:

1. Double a number plus five units.                      a) $x - 5$

2. A number decreased by five units.                         b) $2x + 5$

3. One fifth of a number.                                           c) $x^5$

4. A number raised to the fifth power.                           d) $\frac{x}{5}$

Translate these word phrases into algebraic expressions using a single variable $x$:

1. a) The sum of a number and its half.
2. b) Three times the result of subtracting five from a number.
3. c) The square of the number preceding a given number.
4. d) The difference between double a number and the next number.

Translate the following properties regarding integers into algebraic language:

1. The square of the sum of any two numbers, $a$ and $b$, is equal to the sum of their squares plus double their product.

2. The sum of the squares of three consecutive numbers, $n$ being the smallest of them.

3. The difference between the cube of a number $x$ and the square of its half.

4. The quotient of the sum of a number and its reciprocal, and three times that number.

Copy and complete the following table by identifying the components of these monomials:

From the following list of monomials, identify and group those that are like terms:

$5x^2$, $3y$, $-2x$, $\frac{1}{4}x^2$, $8y$, $x^2$, $-5y$, $10x$

Calculate the numerical value of the following expressions for the given values:

1. a) $3x + 10$ for $x = 4$
2. b) $x^2 - 5$ for $x = 3$
3. c) $2(a + b)$ for $a = 5$ and $b = 2$
4. d) $\frac{12}{x}$ for $x = 6$

Analyse the following algebraic expressions and complete the required information in each case:

1. In the expression $7x^2y - 5x + 3$: How many terms does it have? What is the constant term?

2. Given the monomial $-\frac{2}{3}ab^4$: State its coefficient, its literal part and its total degree.

In a video game store, the cost of renting a console is a fixed fee of $200$ pesos plus $40$ pesos per hour of use.

1. a) Write the algebraic expression for the total cost for $h$ hours.
2. b) How much will Lucía pay if she rents the console for $3$ hours?

On a video game streaming platform, the basic monthly fee is $240$ pesos. In addition, every time you want to play a "Premium" title outside the catalogue, you must pay an extra $70$ pesos.

Write the algebraic expression that represents the total monthly cost a user will pay if they decide to play $x$ "Premium" titles.

A recycling center awards "eco points" based on the material handed in:

• $5$ points for each kilogram of glass ($v$).

• $3$ points for each kilogram of paper ($p$).

• $10$ fixed points for each visit to the center.

Write the formula for the total points $P$ a person receives in one visit. Once written, calculate how many points a student will receive if they hand in $4$ kg of glass and $8$ kg of paper.

Iratxe is analysing the growth of her followers on a social network. She started the month with $x$ followers. During the first fortnight, her number of followers tripled. However, in the second fortnight, she lost $150$ followers due to a cleanup of inactive accounts.

Write the algebraic expression that represents Iratxe's final number of followers and calculate how many she has if she started the month with $x = 420$.

A streaming platform offers two types of subscription:

• Basic Plan: A fixed fee of $x$ pesos per month plus $10$ pesos for each new release movie rented ($p$).

• Premium Plan: A fixed fee that costs $40\%$ more than the basic plan, but allows unlimited movie streaming at no additional cost.

• Write the algebraic expression for the monthly cost of the Basic Plan.

• Write the algebraic expression for the monthly cost of the Premium Plan.

• Write the expression that represents the difference in cost between the Premium Plan and the Basic Plan in a month where $p$ movies are rented. For what values of $p$ would the Basic Plan be more expensive than the Premium Plan? (State only the condition).

A courier company charges its customers based on the weight of the packages and the distance of the delivery. The base fee is $B$ pesos. To this amount is added:

• $50$ pesos for each kilogram of weight ($w$).

• $3$ pesos for each kilometer of distance ($d$).

If the customer is "Premium", a discount of $15\%$ is applied to the total delivery price. Write the algebraic expression that determines the total cost of sending a package for a "Premium" customer in terms of $B$, $w$ and $d$.

Find the algebraic expression that represents the perimeter of the following square whose side length is $x$ meters:

Find the algebraic expression that represents the area of the following rectangle, knowing that its base is $5$ units longer than its height ($x$):

Find the algebraic expression for the perimetre of the following figure, which represents the floor plan of a small rectangular play area.

A cardboard shipping box has a square base with side length $x$ and a height that is $5$ cm greater than the side length of the base.

Draw a sketch of the rectangular prism and write the simplified algebraic expression that represents the volume of the box in terms of $x$.

On a square steel sheet with side length $x$ centimeters, a worker must cut out four identical circles, one at each corner, to manufacture an industrial part. The radius of each circle is $r$ centimeters.

1. Write the algebraic expression for the remaining area of the sheet after making the four cut-outs (use $\pi$ as a symbol).

2. Knowing that the radius of the circles is a quarter of the side length of the sheet, that is $r = \frac{x}{4}$, simplify the previous expression so that it depends only on the variable $x$.

A cardboard box without a lid is to be designed from a rectangular sheet of card measuring $50$ cm long by $30$ cm wide. To do this, a square with side length $x$ is cut out from each of the corners and the flaps are folded up.

Write the simplified algebraic expression that represents:

1. The area of the base of the box in terms of $x$.

2. The total volume of the box in terms of $x$.

Observe the following sequence of patterns made with dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any pattern $n$.

A municipality wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel requires $3$ horizontal slats. To support the structure, a post is placed at the beginning of the fence and another at the end of each panel.

1. Write an algebraic expression to calculate the total number of horizontal slats needed for a fence with $n$ panels.

2. Write an algebraic expression for the total number of posts needed for a fence with $n$ panels.

3. If each post costs $p$ pesos and each horizontal slat costs $l$ pesos, write the expression for the total cost of materials for a fence with $n$ panels.

Observe the following sequence of patterns made of small hexagons. The first pattern has a central hexagon surrounded by six others. In each step, a new layer of hexagons is added around the previous pattern.

1. Determine how many hexagons layer number $n$ has (without counting those in the inner layers).

2. Find the algebraic expression that represents the total number of hexagons that form the complete pattern when it has $n$ layers around the central hexagon.

(Hint: The number of hexagons added in each layer follows a sequence).

Three siblings, Marcos, Lucía, and Adrián, decide to save money to buy a gift. We know that:

• Lucía has saved $x$ pesos.

• Marcos has saved $200$ pesos less than three times what Lucía has.

• Adrián has saved half of the sum of what his other two siblings have together.

Express in terms of $x$:

1. The money Marcos has.

2. The money Adrián has (simplify the expression).

3. The total money saved by the three siblings.