Area Questions and Answers

Answer: B

The area must stay fixed, so the two factors have to multiply to 1.

\[ 1.25 \times (1 - d) = 1 \quad\Longrightarrow\quad 1 - d = \frac{1}{1.25} = 0.8 \]

\[ d = 0.2 = 20\% \]

A 25% rise is undone by a 20%, not a 25%, fall, because the reduction applies to the larger figure. The general rule is

\[ d = \frac{x}{100 + x} \times 100 = \frac{25}{125} \times 100 = 20\% \]

Answer: C

Turn each perimeter into a side, square it, add, then work back.

\[ \text{sides} = \frac{24}{4},\; \frac{32}{4},\; \frac{40}{4},\; \frac{76}{4},\; \frac{80}{4} = 6,\; 8,\; 10,\; 19,\; 20 \]

\[ \text{total area} = 36 + 64 + 100 + 361 + 400 = 961 \text{ cm}^{2} \]

\[ \text{side of new square} = \sqrt{961} = 31 \text{ cm} \]

\[ \text{perimeter} = 4 \times 31 = 124 \text{ cm} \]

Perimeters do not simply add: their sum is 252 cm, twice the answer, because area grows with the square of the side.

Answer: D

Use the equilateral-triangle area formula and solve for the side.

\[ A = \frac{\sqrt{3}}{4}a^{2} = 4\sqrt{3} \]

\[ a^{2} = \frac{4\sqrt{3} \times 4}{\sqrt{3}} = 16 \]

\[ a = 4 \text{ cm} \]

The \( \sqrt{3} \) cancels from both sides, which is why the answer comes out a whole number.

Answer: B

Both dimensions grow, so multiply the two factors.

\[ \frac{A'}{A} = (1 + 0.10)(1 + 0.25) = 1.10 \times 1.25 = 1.375 \]

\[ \text{increase} = 1.375 - 1 = 0.375 = 37.5\% \]

Adding the percentages gives 35%; the extra 2.5% is the cross term \( 0.10 \times 0.25 \), the part of the rectangle that gains in both directions at once.

Answer: A

We know perimeter of square = 4(side)
So Side of first square = 40/4 = 10 cm
Side of second square = 32/4 = 8 cm
Area of third Square = 10*10 - 8*8
= 36 cm
So side of third square = 6 (because area of square = side*side)
Perimeter=4*Side=4*6=24 cm

Answer: D

Place the centre at the origin with \( A = (0,\,5) \). By symmetry, equal chords \( AB \) and \( AC \) put B and C at the same height, so \( BC \) is horizontal.

For \( B = (x,\,y) \) on the circle with \( AB = 6 \):

\[ x^{2} + y^{2} = 25, \qquad x^{2} + (y - 5)^{2} = 36 \]

Subtracting:

\[ -10y + 25 = 11 \quad\Longrightarrow\quad y = 1.4 \]

\[ x^{2} = 25 - 1.96 = 23.04 \quad\Longrightarrow\quad x = 4.8 \]

So \( B = (4.8,\,1.4) \) and \( C = (-4.8,\,1.4) \):

\[ BC = 2 \times 4.8 = 9.6 = \frac{48}{5} \text{ cm} \]

None of the listed lengths is \( \tfrac{48}{5} \), so the answer is none of these. The \( \tfrac{24}{5} \) on offer is only half of \( BC \) - the distance from B to the axis of symmetry.

Answer: A

Convert the mixed numbers to decimals and divide the areas.

\[ \text{stone} = 2.5 \times 2 = 5 \text{ m}^{2} \]

\[ \text{courtyard} = 30 \times 16.5 = 495 \text{ m}^{2} \]

\[ n = \frac{495}{5} = 99 \]

The division is exact, so the stones tile the courtyard with no cutting - which is what makes the volume comparison legitimate here.

Answer: B

Express the breadth in terms of the length and use the perimeter.

\[ b = 0.6l \quad\Longrightarrow\quad 2(l + 0.6l) = 800 \]

\[ 2(1.6l) = 800 \quad\Longrightarrow\quad l = 250 \text{ m} \]

\[ b = 0.6 \times 250 = 150 \text{ m} \]

\[ A = 250 \times 150 = 37{,}500 \text{ m}^{2} \]

Reading the 60% as applying to the length rather than the breadth reverses the two and halves the area to 18,750 - the first distractor.

Answer: B

From \( d = a\sqrt{2} \):

\[ a = \frac{8\sqrt{2}}{\sqrt{2}} = 8 \]

\[ \text{perimeter} = 4a = 32 \text{ m} \]

The \( \sqrt{2} \) in the given diagonal is a deliberate signal that the side is a whole number.

Answer: C

A sector is a triangle-like wedge whose "base" is its arc and whose "height" is the radius:

\[ A = \frac{1}{2} \times r \times \text{arc length} \]

\[ = \frac{1}{2} \times 5 \times 3.5 = 8.75 \text{ cm}^{2} \]

This form needs no angle at all. It follows from \( A = \tfrac{\theta}{2\pi}\pi r^{2} \) and \( \text{arc} = r\theta \), which combine to give exactly \( \tfrac{1}{2}r \times \text{arc} \).