Introduction To Ratio

I. Introduction

Ratios are useful for comparing two or more quantities easily. If you have $1000$ candies and your friend has $500$ candies, it is very simple to say, you have twice as much candies as your friend.

Using ratio, you could say:

$Number\ of\ candies\ you\ have\ \underline{is\ to}\ number\ of\ candies\ your\ friend\ has\ is\ two\ \underline{is\ to}\ one$.

Which is other words means that for every $1$ candy that your friend has, you have $2$ candies. Mathematically, the symbol $:$ is used to denote $\underline{is\ to}$.

Let us say you have $75$ pencils, while your friend has $100$ pencils. So we can say that for every $3$ pencils that you have, your friend has four pencils.

Ratios are, in a sense, very similar to fractions, and in many places ratios are represented as fractions. If you take the above example, the ratio of your pencils and your friends pencils is:

$\dfrac{\cancel{75} \raise{0.2em}{\cancel{15} \raise{0.2em}{3}}}{\cancel{100} \lower{0.2em}{\cancel{20} \lower{0.2em}{4}}} = \dfrac{3}{4}$

Like fractions, ratios are most commonly denoted for quantities having the same unit.

One very important thing to remember is that ratio can be of more than two quantities as well.

So, if $A$ has $45$ chocolates, $B$ has $75$ chocolates and $C$ has $90$ chocolates, we can say that

$A:B:C = 45:75:90$, we cancel out the GCD/HCF $15$ from all three numbers, and we get:

$A:B:C = 3:5:6$

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Let us try to see a few examples to understand this better.

Let us say, we have a triangle whose angles are in the ratio of $3:5:10$, what are the angles.

Let us say the $3$ angles had a common factor $x$ which was cancelled out to give us $3:5:10$, we can say that the angles have values of $3x$, $5x$ and $10x$. But we also know that the sum of all $3$ angles of a triangle is $180^\circ$.

$\texttip{\therefore}{therefore} 3x + 5x + 10x = 180$

$\texttip{\therefore}{therefore} 18x = 180$

$\texttip{\therefore}{therefore} x = \dfrac{180}{18} = 10$

Therefore the 3 angles of our triangle would be $3 \times 10 = 30^\circ$, $5 \times 10 = 50^\circ$ and $10 \times 10 = 100^\circ$

Let us take one more example:

We divide $100$ candies between four friends $A$, $B$, $C$ and $D$ in the ratio of $5:4:4:7$. How many did each receive?

Like before, we will assume that $x$ was the common factor that was cancelled out to get our ratio. So, we can say that $A$ received $5x$, $B$ received $4x$, $C$ received $4x$ and $D$ received $7x$ candies. We know that in all, they received $100$ candies.

Therefore:

$5x + 4x + 4x + 7x = 100$

$\texttip{\therefore}{therefore} 20x = 100$

$\texttip{\therefore}{therefore} x = \dfrac{100}{20} = 5$

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So,

$A$ received $5x = 5 \times 5 = 25$ candies,

$B$ received $4x = 4 \times 5 = 20$ candies,

$C$ received $4x = 4 \times 5 = 20$ candies, and

$D$ received $7x = 7 \times 5 = 35$ candies.

You can validate that $25 + 20 + 20 + 35 = 100$

Let's say Martina and Chris' maths score are in the ratio $11:14$. Chris scored $15$ more than Martina. How much each of them score?

Let us assume, that the common factor in the ratio was $x$.

Therefore, Chris scored $14x$ and Martina scored $11x$. But we know that Chris scored $15$ more than Martina.

Therefore:

$14x - 11x = 15$

$\texttip{\therefore}{therefore} 3x = 15$,

$\texttip{\therefore}{therefore} x = \dfrac{15}{3} = 5$

Hence, Chris scored $14x = 14 \times 5 = 70$ and Martina scored $11x = 11 \times 5 = 55$.

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Now let us try the following problem before we move onto the next section:

--------- Reference to question: 62c0f562-f8a9-4379-b0fe-810201e68d6a ---------

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II. Relationship With Fractions

Ratios are a quite similar to fractions in some sense. While fraction tell us what part of a whole we are interested in, ratio gives us a comparison of two quantities directly. While one fraction tells us about one item at a time, ratio can tell us about more than one item at the same time.

For example, we can say that $A:B:C = 2:3:5$. Converting this ratio to fraction, we can say that $A$ constitutes $\dfrac{2}{10} = \dfrac{1}{5}$ of the whole. But this fraction does not tell us the amount of $B$ or $C$ individually.

That's why ratios are considered a better suited for expressing a comparative view of more than two things using just one value, while fractions are better suited when it comes to giving an estimate of one quantity as a part of a whole.

Taking our example of $A:B:C = 2:3:5$ it is possible to express each one of them as a separate fraction, like $A$ constitutes $\dfrac{1}{5}$ of the whole, $B$ constitutes $\dfrac{3}{10}$ of the whole and $C$ constitutes $\dfrac{1}{2}$ of the whole. However, given any one these fractions we cannot re-construct the complete information about all three quantities. We need two fractions to re-construct the entire information.

III. Converting To Fractions

Now we will see how to convert a given ratio to fractions to represent each of its constituents.

Suppose we are given that an alloy is made by mixing iron, copper and aluminium in the ratio $4:3:9$. Now, if we want to find the fraction of iron in the alloy, all we need to do is to take the value corresponding to iron, and divide it by the sum of all three components.

Therefore, the fraction of iron is $\dfrac{4}{4 + 3 + 9} = \dfrac{4}{16} = \dfrac{1}{4}$

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Let's understand this a little bit better by looking at the detailed working.

Any amount of the alloy you take, it will contain $4x$, $3x$ and $9x$ of iron, copper and aluminium respectively. Therefore, the fraction of iron in that amount of alloy is given by:

$\dfrac{4x}{4x + 3x + 9x} = \dfrac{4x}{16x} = \dfrac{4}{16} = \dfrac{1}{4}$

Now we can try this question:

--------- Reference to question: d62702b4-9cad-4202-85c3-19313f64fcb1 ---------