Algebraic language
A courier company charges a fixed rate $B$ plus $2{,}5$ € for each kilo. How much does it cost to send a parcel of $w$ kilos? The answer, $B + 2{,}5w$, works for any parcel, without having to redo the calculation each time. That is what mastering algebraic language gives you.
Multi-step sentences with several letters
The sentences in this course mix several letters and several steps. Translate in parts, working from the inside out.
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: $2,5$…
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:
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$2,5$ euros for each kilogram of weight ($w$).
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$0,15$ 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 $d$.
Matching sentences and expressions
You also need to recognise, the other way round, which sentence corresponds to an expression that is already written. It is the same skill, but in the opposite direction.
| Expression | In everyday language |
|---|---|
| $2x + 5$ | Twice a number plus five |
| $x - 5$ | A number decreased by five |
| $\dfrac{x}{5}$ | One fifth of a number |
| $x^5$ | A number raised to the fifth power |
Match each phrase with its corresponding algebraic expression: 1. Double a number plus five units.…
Match each phrase with its corresponding algebraic expression:
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Double a number plus five units. a) $x - 5$
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A number decreased by five units. b) $2x + 5$
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One fifth of a number. c) $x^5$
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A number raised to the fifth power. d) $\frac{x}{5}$
Tariff problems: fixed plus variable
Many real tariffs combine a fixed part (which does not change) and a variable part (which depends on a quantity). They are always translated in the same way: $\text{fixed} + \text{variable} \cdot \text{quantity}$.
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. a) Write the algebraic expression for the total cost for…
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.
- a) Write the algebraic expression for the total cost for $h$ hours.
- b) How much will Ainhoa pay if she rents the console for $3$ hours?
Consecutive, even and odd numbers
To describe numbers that are related to each other, the first one is named with a letter and the others are written in terms of it:
| Description | Expression |
|---|---|
| Any number | $n$ |
| The next one | $n + 1$ |
| Two consecutive numbers | $n,\ n+1$ |
| Three consecutive numbers | $n,\ n+1,\ n+2$ |
| An even number | $2n$ |
| An odd number | $2n + 1$ |
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…
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.
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Write the algebraic expression for the remaining area of the sheet after making the four cut-outs (use $\pi$ as a symbol).
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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 $x$.
Proving with algebra
With specific numbers you only check one case. With a letter you prove that something is true always.
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…
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:
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The area of the base of the box in terms of $x$.
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The total volume of the box in terms of $x$.