# Tutor profile: Alison B.

## Questions

### Subject: Geometry

The perimeter of a right-angled triangle is 72 cm. The lengths of its sides are in the ratio 3 : 4 : 5 Work out the area of the triangle.

The equation for the area of a triangle is $$ area = \frac{1}{2} \times base \times height $$ So, first we need to know the length of the base and the height of the triangle. We know that the perimeter of the triangle is 72cm, and the ratio of the sides is 3 : 4 : 5. To work out the length of the three sides, we add up each part of the ratio (3 + 4 + 5 = 12), and divide the perimeter by it $$ (72 \div 12 = 6). $$ Then we multiply each part of the ratio by 6, so we have the sides being 18 : 24 : 30. We can check that these total 72. Now, the hypotenuse of the triangle is always the longest side; and since it is a right angled triangle we know that one of the sides will be the base and one will be the height. Note that because the original equation has us multiplying the two values, it does not matter which is which. So $$ base = 3 \times 6 = 18 $$ and $$ height = 4 \times 6 = 24 $$, or vice versa. Putting these back into the equation, we have $$ area = \frac{1}{2} \times 18 \times 24 = 216cm^{2} $$

### Subject: Basic Math

In a village: - the number of houses and the number of flats are in the ratio 7 : 4 - the number of flats and the number of bungalows are in the ratio 8 : 5 - There are 50 bungalows in the village. How many houses are there in the village?

Firstly, we can use the number of bungalows in the village, and the ratio between flats and bungalows, to work out the number of flats in the village. The ratio of flats to bungalows is 8 : 5. So, we divide the number of bungalows by 5, and then multiply by 8: $$ 50\div5 = 10 $$ $$ 10 \times 8 = 80 $$ Now that we know the number of flats in the village, we can use the ratio of houses to flats to work out the number of houses. Here, the ratio of houses of flats is 7 : 4, so we divide the number of flats by 4 and then multiply by 7. So we have: $$ 80 \div 4 = 20 $$ $$ 20 \times 7 = 140 $$ There are 140 houses in the village.

### Subject: Algebra

Simplify fully $$ 2 (x^2 + 4) – 2 (x^2 – 2) $$

First, we multiply out the brackets: $$2x^2 + 2 \times 4 - 2 x^2 - 2 (-2)$$ = $$2x^2 + 8 - 2 x^2 +4$$ Note that in the last term, the double negative makes a positive. Then, we collect the 'like terms'. This means adding up the constant terms, then the $$ x^2 $$ terms, then the $$ x ^3 $$ terms, and so on. the Here, the $$ x^2 $$ terms cancel out, so we are left with just the constant terms: $$ = 8 + 4 = 12 $$

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