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

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

If (π‘₯ + 𝑦)/28 = (π‘₯ βˆ’ 𝑦)/2 = (π‘₯ + 𝑦 βˆ’ 𝑧)/4, find the ratio π‘₯ : 𝑦 : 𝑧 in its simplest form.

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Video Transcript

If π‘₯ plus 𝑦 over 28 is equal to π‘₯ minus 𝑦 over two which is equal to π‘₯ plus 𝑦 minus 𝑧 over four, find the ratio π‘₯ to 𝑦 to 𝑧 in its simplest form.

A ratio is said to be in its simplest form where all our values are integers with no common factor apart from one. Let’s begin by considering the first two terms in our equation, π‘₯ plus 𝑦 over 28 is equal to π‘₯ minus 𝑦 over two. We can begin here by multiplying both sides of the equation by 28. On the left-hand side, the twenty eights cancel, and we are left with π‘₯ plus 𝑦. On the right-hand side, 28 divided by two is equal to 14, so we have 14 multiplied by π‘₯ minus 𝑦. Distributing the parentheses or expanding the brackets here gives us 14π‘₯ minus 14𝑦.

At this stage, we can subtract π‘₯ and add 14𝑦 to both sides of the equation. This gives us 15𝑦 is equal to 13π‘₯. Dividing both sides by 15, we have 𝑦 is equal to thirteen fifteenths π‘₯. Let’s now consider the second and third terms of the equation, π‘₯ minus 𝑦 over two is equal to π‘₯ plus 𝑦 minus 𝑧 over four. This time, we begin by multiplying both sides of the equation by four. On the left-hand side, we have two multiplied by π‘₯ minus 𝑦. On the right-hand side, we have π‘₯ plus 𝑦 minus 𝑧.

Once again, we can distribute the parentheses, giving us two π‘₯ minus two 𝑦. Rearranging this equation by adding 𝑧 to both sides, adding two 𝑦 to both sides, and subtracting two π‘₯ from both sides gives us 𝑧 is equal to three 𝑦 minus π‘₯. We already know that 𝑦 is equal to thirteen fifteenths of π‘₯. This means that 𝑧 is equal to three multiplied by thirteen fifteenths of π‘₯ minus π‘₯. 13 minus thirteen fifteenths is equal to thirty-nine fifteenths or 39 over 15. We can then subtract π‘₯ from 39 over 15π‘₯. This gives us 𝑧 is equal to 24 over 15π‘₯.

We now have two equations, with both 𝑦 and 𝑧 written in terms of π‘₯. We will now clear some space so we can find the ratio π‘₯ to 𝑦 to 𝑧. In order to write the ratio in the form π‘₯ to 𝑦 to 𝑧, we can replace 𝑦 with 13 over 15 π‘₯ and 𝑧 with 24 over 15 π‘₯. All three of our values are now in terms of the same variable π‘₯. This means that we can divide each of the ratios by π‘₯. The ratio π‘₯ to 𝑦 to 𝑧 is one to 13 over 15 to 24 over 15. This is not in its simplest form as our second and third values are not integers. We need to multiply each of the ratios by 15 to get rid of the denominators. This gives us 15, 13, and 24, respectively. The ratio π‘₯ to 𝑦 to 𝑧 in its simplest form is 15 to 13 to 24.

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