Exercises: Algebraic language

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

1. a) The number of followers an influencer has, $x$, increased by $150$.
2. b) The price of a cinema ticket, $p$, reduced by $2$ euros.
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.

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

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

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 $x = 6$

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

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) $\frac{x}{5}$

In a video game shop, the cost of renting a console is a fixed fee of $10$ euros plus $2$ euros per hour of use.

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

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

Cian rents a shared electric scooter in Dublin. The company charges a fee of $1$ € for the initial unlock plus $0.25$ € for each minute of the ride. To calculate the cost in euros of a ride of $m$ minutes, Cian writes the expression $(1 + 0.25) \times m$ in his notebook.

1. Identify the conceptual error that Cian made when setting up this expression.

2. Explain in your own words why the fixed term should never be multiplied by the time variable in this type of pricing.

3. Write the correct algebraic expression to calculate the total cost of the ride as a function of the minutes $m$.

In an Irish town, the monthly domestic water bill consists of a fixed maintenance and sewage fee of $11.50$ €, to which a cost of $1.30$ € is added for each cubic metre ($m^3$) of water consumed.

1. Write the algebraic expression that allows you to calculate the total amount of the monthly bill for a household that has consumed $x$ cubic metres of water.

2. If a family consumes $14\text{ m}^3$ during the month of May, calculate the total amount that will appear on their bill.

3. If a home remains unoccupied during the holidays and its consumption is $0\text{ m}^3$, how much will they pay that month? Explain how this result is deduced from your algebraic expression.

On a video game streaming platform, the basic monthly fee is $12$ €. In addition, every time you want to play a "Premium" title outside the catalogue, you must pay an extra $3.50$ €.

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

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

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 recycling centre 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 centre.

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.

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.

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 $x = 420$.

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.

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 cuboid and write the simplified algebraic expression that represents the volume of the box in terms of $x$.

You can do it on paper and send a photo with your answer.

A delivery van operating in a provincial capital has a daily operating cost modelled by the expression $C = 16.50 + 0.12 \times k$, where $C$ is the total daily expense in euros and $k$ is the distance travelled in kilometres ($16.50$ € for daily fixed insurance and taxes, plus $0.12$ € per kilometre for fuel and wear and tear).

1. Before carrying out the exact calculation, make a mental estimate of the total cost for a route of $195\text{ km}$. To do this, round $16.50$ € to $17$ € and $0.12$ € to $0.10$ € (or use another justified rounding that makes mental calculation easier). Show the estimated operation you performed.

2. Calculate the exact actual cost by substituting $k = 195$ into the original algebraic expression.

3. Compare the exact value with your previous estimate: calculate the difference in euros and argue whether the estimate was reasonable and sufficient to forecast the day's expenses.

You want to invent a mathematical guessing game for your classmates. The game must mathematically lead to the following algebraic expression:

$$\frac{2 \times (x + 5) - 4}{2}$$

where $x$ is the secret number your classmate initially chooses.

1. Write in order the step-by-step verbal instructions you must dictate to your classmate so that they build this expression (starting with "Think of any integer...").

2. Expand and simplify the algebraic expression as much as possible by applying the distributive property and combining like terms.

3. Based on the simplified expression, explain what the "trick" is: if your classmate performs the operations silently and tells you out loud that they obtained $19$, what instant mental calculation must you do to tell them which number $x$ they thought of at the beginning?

A local council 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 $p$ euros and each horizontal slat costs $l$ euros, write the expression for the total cost of materials for a fence with $n$ panels.

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

• $2.5$ euros for each kilogram of weight ($w$).

• $0.15$ euros for each kilometre 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 parcel for a "Premium" customer in terms of $B$, $w$ and $d$.

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

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 $x$.

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).

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.

Three siblings, Cian, Aoife and Darragh, decide to save money to buy a present. We know that:

• Aoife has saved $x$ euros.

• Cian has saved $10$ euros less than three times what Aoife has.

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

Express in terms of $x$:

1. The money Cian has.

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

3. The total money saved by the three siblings.

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 $x$.

A streaming platform offers two types of subscription:

• Basic Plan: A fixed fee of $x$ euros per month plus $0.50$ euros 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).

Two 2nd Year students, Cian and Aoife, want to prove in a general way that "the sum of three consecutive integers is always divisible by $3$". Each proposes a different algebraic approach:

• Cian's method: he calls the first number $n$, the second $n + 1$, and the third $n + 2$.

• Aoife's method: she calls the central number $n$, the previous one $n - 1$, and the following one $n + 1$.

• Expand and simplify the sum proposed by Cian. Justify why the expression obtained proves that the result is always a multiple of $3$, regardless of the value of the integer $n$.

• Expand and simplify the sum proposed by Aoife. Justify why it also proves the same property.

• Compare both methods: which of the two is more direct and elegant when simplifying the numerical terms? Justify your answer.

• A classmate says: "Why complicate things with letters? Just by checking that $4+5+6=15$ and that $10+11+12=33$, you can already see that the rule works". Refute their argument by explaining what the essential function of algebra is compared to simply testing with numerical examples.

In the local school futsal tournament, $n$ teams participate. In the preliminary phase, all teams must play against each other exactly once (single round-robin system). The organisation uses the algebraic expression $P = \frac{n \times (n - 1)}{2}$ to calculate the total number of matches that must be scheduled.

1. Interpret the logical structure of the formula: explain why the number of teams $n$ is multiplied by $(n - 1)$, and why it is mandatory to divide that product by $2$.

2. Since the number of matches must necessarily be an integer for any number of teams $n$ ($n \ge 2$), prove algebraically that $P$ always yields an exact integer result by studying the two possible cases separately:

3. Case A: The number of teams is even ($n = 2k$, where $k$ is a natural number).

4. Case B: The number of teams is odd ($n = 2k + 1$, where $k$ is a natural number).

(Substitute each form into the formula for $P$ and simplify the fraction).

1. If the municipal sports centre only has available hours to play a maximum of $20$ matches over the weekend, determine using the algebraic formula whether it is possible to admit a tournament of $7$ teams.