Video: Solving Proportion Equations to Find the Values of Unknowns in Ratios

If 𝑎 = 1/2 𝑏 and 𝑐 = 𝑎 + 𝑏, then express the ratio 𝑎 : 𝑏 : 𝑐 in its simplest form.

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Video Transcript

If 𝑎 is equal to a half 𝑏 and 𝑐 is equal to 𝑎 plus 𝑏, then express the ratio 𝑎 to 𝑏 to 𝑐 in its simplest form.

A ratio is said to be in its simplest form when all of our values are integers with no common factor apart from one. In this question, we need to write the ratio in the form 𝑎 to 𝑏 to 𝑐. We know that 𝑎 is equal to a half of 𝑏. If we multiply both sides of this equation by two, we get two 𝑎 is equal to 𝑏 or 𝑏 is equal to two 𝑎. We will call this equation one.

We are also told that 𝑐 is equal to 𝑎 plus 𝑏. As 𝑏 is equal to two 𝑎, we can substitute this into the equation. This means that 𝑐 is equal to 𝑎 plus two 𝑎. 𝑐 is therefore equal to three 𝑎. We will call this equation two.

As previously mentioned, we need to write the ratio in the form 𝑎 to 𝑏 to 𝑐. We know that 𝑏 is equal to two 𝑎 from equation one. From equation two, we know that 𝑐 is equal to three 𝑎. The ratio can therefore be rewritten as 𝑎 to two 𝑎 to three 𝑎. All of our terms have a common variable 𝑎. This means we can divide them all by 𝑎.

The ratio of 𝑎 to 𝑏 to 𝑐 is therefore equal to one to two to three. As these three values have no common factor apart from one, the ratio is in its simplest form.