# Daily Maths Revision – Week 15 Walkthrough

If you’re struggling with this week’s questions, this walkthrough should help you get the skills you need.

This week, we’re exploring:

If it’s a formula you need, here are the ones that are relevant this week. In 2023, you are given the formula for the area of a trapezium. However, you’re not given the formula for speed so you will need to learn it.

Area of a trapezium:

Where 𝒂 and 𝒃 are the lengths of the parallel sides and 𝒉 is their perpendicular separation: $area = \frac{1}{2}(a + b)h$

Speed: $speed = \frac{distance}{time}$

### Question 1: The Area of a Trapezium

To find the area of a trapezium, we add together the parallel sides then multiply the result by the perpendicular height and divide by 2. In other words, if we call the parallel sides 𝑎 and 𝑏: $area = \frac{1}{2}(a + b)h$

Find the area of the trapezium below. \begin{aligned} area &= \frac{1}{2}(a + b)h \\ &= \frac{1}{2}(10 + 17) \times 9 \\ &= 121.5cm^2 \end{aligned}

### Question 2: Multiplying Vectors by Scalars

When multiplying vectors by a scalar, we simply multiply each component by the scalar.

Given $\bold{a} = \begin{pmatrix} 2 \\ -6 \end{pmatrix}$, find the vector $-\frac{1}{3}\bold{a}$. \begin{aligned} -\frac{1}{3}\bold{a} &= -\frac{1}{3} \times \begin{pmatrix} 2 \\ -6 \end{pmatrix} \\ &= \begin{pmatrix} -\frac{1}{3}\times2 \\ -\frac{1}{3}\times-6 \end{pmatrix} \\ &= \begin{pmatrix} -\frac{2}{3} \\ 2 \end{pmatrix} \end{aligned}

### Question 3: Solving Equations

Solving equations means finding the specific value the satisfies that equation. Usually, this means finding the value of a particular letter.

Let’s solve: $\huge{\boldsymbol{7x - 8 = 2(x + 1)}}$

We need to start by expanding the brackets on the right-hand side. The easiest way to do this is by using the grid method – this was covered in Week 1. We place the term on the outside (2) on the left and the terms inside the brackets across the top. Then, we multiply the term on the left by the term in each column.

So, our equation is now: $7x - 8 = 2x + 2$

Now, we want to solve the equation by collecting all the variables (the terms with letters) on one side: \begin{aligned} & &7x-8&=2x+2 \\&-2x&&&-2x\\&&5x-8&=2\\&+8&&&+8\\&&5x&=10\\&\div5&&&\div5\\&&x&=2\end{aligned}

### Question 4: Average Speed

To find the average speed, we need to divide the total distance by the total time. You can revise how to find the speed when you have these values by looking back at week 2.

Jonathon travels 500m at 8m/s and then travels at 4m/s for 30 seconds. What was Jonathon’s average speed? Give your answer to 1 decimal place.

First, we need to find the time taken for the first part of the journey. We can use a formula triangle to help us remember the different operations.

We are looking for time, so we divide the distance by the speed.

time = 500 ÷ 8
= 62.5s

Next, we need the distance travelled in the second part of the journey. To find the distance, we multiply the speed by the time.

distance = 4 × 30
= 120m

Finally, we divide the total distance (120 + 500) by the total time (62.5 + 30):

Average speed = (120 + 500) ÷ (62.5 + 30)
= 6.7m/s (1d.p.)

### Question 5: Experimental Probability

Experimental probability, sometimes called relative frequency, is a method to estimate the probability of an event happening when we can’t know the exact value. For example, the probability of a drawing pin landing “pin up”. Depending on the style and make of the pin, we can’t work this out exactly, but we can estimate the probability by trying it again and again. The more times we try it, the more accurate the estimate will be.

A drawing pin is thrown 500 times. It lands “pin up” 375 times. Estimate the probability that, when thrown again, it will land “pin up”.

The experimental probably is simply the number of successes divided by the number of trials. \begin{aligned} \text{P(pin up)} &= \frac{\text{number of successes}}{\text{number of trials}}\\ &= \frac{375}{500} \\ &= \boldsymbol{0.75} \end{aligned}

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