Key Stage 3

Estimating Percentages II

Lesson

We've already learnt to estimate percentages of pictures that can give us a fraction out of $5$5, $10$10 or $100$100, but what about something that's not so easy to work with? Let's have a look at the following example:

What percentage of the lights are turned on?

Well, to approach any difficult percentage question it is always useful to first think of it as a fraction. Here we have $7$7 lights in total, and $4$4 are on, so the fraction of lights that are on is $\frac{4}{7}$47.

Now we can use what we've learnt about percentages and fractions previously to transform it into a percentage by multiplying it by $100%$100%.

Therefore the answer is $\frac{4}{7}\times100%=57.14%$47×100%=57.14% when rounded to two decimal places.

All for One and One for All!

We know that percentages and fractions are used to describe parts of wholes, and that whole is represented as $1$1 or $\frac{x}{x}$xx (where $x$x is any number) in fractional form, and $100%$100% as a percentage. This is important when we come to talk about complementary fractions and percentages, which are fractions and percentages that add together to give a whole.

Let's take a look at the above example again: $\frac{4}{7}$47 of the lights are on, but how many are off then? This seems very simple, and the answer is $\frac{3}{7}$37.

Can you see that $\frac{4}{7}$47 and $\frac{3}{7}$37 are complementary because they add to give $\frac{7}{7}=1$77=1?

This is because being on and being off are the only two possibilities for our lights in our diagram there. We can see the whole represented as a round circle in a pie chart below:

In another situation we might have had some lights on, some off, and some flickering. In that case we would have three different complementary fractions describing three different groups of lights.

We can see the same thing when talking about percentages. Let's say we knew that $30%$30% of a class is made up of people who wear glasses, how do we know what percentage don't?

Well, we know that glasses-wearers and non-wearers are the only two possible groups, so together they must make a whole = $100%$100%. That means the percentage of people who don't wear glasses is $100%-30%=70%$100%−30%=70%.

Worked Examples

QUESTION 1

Which point on the line is closest to $56%$56%?

A horizontal number line ranging from 0% on the far left to 100% on the far right. This number line is divided into equal segments, each representing a 10% interval, but only the 0% and 100% markers are labeled. Four arrows, labeled A, B, C, and D, are positioned above the line, each pointing downwards to specific intervals on the scale. (The arrow labeled A points between 20% and 30%, B's arrow points between 30% and 40%, C's arrow is between 50% and 60%, and D's arrow is between 80% and 90%. The location of the arrows is not explicitly given, so just use them as hints for the students. Remember that the students are asked which of the letters is closest to the given number.)

$A$A
A
$C$C
B
$D$D
C
$B$B
D

QUESTION 2

$Ellie$ bought a $454$454 mL drink that claimed to be orange juice. In the ingredients list it said that orange juice made up $17%$17% of the drink. To estimate the amount of orange juice in the drink, which of the following would give the closest answer?

$10%\times454$10%×454
A
$20%\times454$20%×454
B
$10%\times400$10%×400
C

QUESTION 3

In a census, people are asked their gender and age. The graph shows the results: the percentage of females and males in each age group.

To the nearest $1%$1%, what percentage of females are between $5$5 and $9$9 years of age?
$7%$7%
A
$2%$2%
B
$11%$11%
C
To the nearest $1%$1%, what percentage of males are between $30$30 and $34$34 years of age?
$7%$7%
A
$4%$4%
B
$2%$2%
C
The percentage of females between the ages of $20$20 and $29$29 is about:
$15%$15%
A
$7%$7%
B
$25%$25%
C
The percentage of males below $20$20 years of age is about:
$15%$15%
A
$10%$10%
B
$30%$30%
C
$50%$50%
D