Solving the toughest 2023 PSLE Maths question on Ratios and Fractions

Mathematics can be a challenging subject, and one of the most formidable aspects for many students is understanding and working with ratios and fractions. In the 2023 PSLE exam, students encountered a particularly tough maths question that put their knowledge to the test. In this article, we'll explore the intricacies of ratios and fractions and provide you with a step-by-step guide to conquer this challenging question.

Understanding Ratios And Fractions

Ratios are a way to compare two or more quantities, expressing their proportional relationship. Fractions, on the other hand, represent a part of a whole. In many maths problems, including the 2023 PSLE question we will discuss below, these concepts intertwine, and a solid grasp of them is essential.

Tackling the complex 2023 PSLE Maths Question on Ratios and Fractions

Question:

In the figure, $\text{ADGE}$ is a rectangle where $\mathrm{AB = BC = CD}$. $\text{CE}$ intersects $\text{BG}$ at $\text{F}$. Given that the area ratio of $\mathrm{\triangle AEC}$ to $\mathrm{\triangle FBC}$ is $8:1$, what fraction of rectangle $\text{ADGE}$ is shaded?

$2023 psle maths question ratios and fractions$

Solution:

Let's look at the question,

We're told that $\mathrm{\triangle AEC : \triangle FBC = 8 : 1}$

So, if the area of $\mathrm{\triangle FBC = 1}$ unit,

Then the area of $\text{AEFB}$ would be $8 - 1 = 7$ units.

Since $\text{ADGE}$ is a rectangle, $\text{AE}$ would be the same height as $\text{GD}$.

This means that,

Area of $\text{AEFB}$ = Area of $\text{GFCD = 7}$ units.

So, if we add the area of $\text{AEFB}$, area of $\text{GFCD}$, and area of $\mathrm{\triangle FBC}$, we get the area of the shaded area.

Therefore, the area of shaded area = $7+7+1=15$ units.

Now, if we just knew the area of rectangle $\text{ADGE}$, we'd be able to find the exact area of the shaded area.

How do we find that?

We know that $\mathrm{AB = BC = CD}$ and that the area of $\mathrm{\triangle AEC : \triangle FBC = 8 : 1}$.

Let's imagine that there was a straight line that went up straight from point $\text{B}$. And let the point at which that line would have touched line $\text{EG}$ be $\text{K}$.

So, the rectangle $\mathrm{KBDG}$ would be $\displaystyle{\frac{2}{3}}$rd of the rectangle $\mathrm{AEGD}$, and $\mathrm{\triangle KGB}$ would be 8 units.

This means that the area of $\displaystyle{\frac{2}{3}}$rd of the rectangle is $8+1+7 = 16$ units.

And $\displaystyle{1 \over 3}$rd of the rectangle would be $\displaystyle{\frac{16}{2}= 8}$ units.

This also means that the area of $\displaystyle{3 \over 3}$ of the rectangle will be $\displaystyle{8 \times 3 = 24}$ units.

We already know that the area of the shaded figure is 15 units.

The fraction of the shaded figure $\displaystyle{={15 \over 24} = {5\over 8}}$.

Answer:

$\displaystyle{\frac{5}{8}}$

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