Video: Pack 4 • Paper 2 • Question 16

Pack 4 • Paper 2 • Question 16

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Video Transcript

Given that the ratio of four 𝑥 minus one to 𝑥 plus two is equal to the ratio of three 𝑥 minus two to 𝑥, find all possible values of 𝑥.

As the two ratios are equal, this can be rewritten as two fractions. Four 𝑥 minus one divided by 𝑥 plus two is equal to three 𝑥 minus two divided by 𝑥. We can prove that this method will work by using some number examples. The ratio five to three is the same as the ratio 10 to six. Therefore, five to three is equal to 10 to six. This can be rewritten as five divided by three or five-thirds is equal to 10 divided by six or ten sixths.

We can see that these two fractions are equivalent as we have multiplied the top and the bottom, the numerator and the denominator, by two. Whatever you do to the top, you must do to the bottom. This little proof shows us that we can rewrite the question as four 𝑥 minus one divided by 𝑥 plus two is equal to three 𝑥 minus two divided by 𝑥. Cross multiplying or multiplying both fractions by 𝑥 and 𝑥 plus two gives us 𝑥 multiplied by four 𝑥 minus one is equal to 𝑥 plus two multiplied by three 𝑥 minus two.

Expanding the single bracket on the left-hand side gives us four 𝑥 squared minus 𝑥 as 𝑥 multiplied by four 𝑥 is four 𝑥 squared and 𝑥 multiplied by negative one is negative 𝑥. We can expand or multiply out the two brackets on the right-hand side using the FOIL method. Multiplying the first terms gives us three 𝑥 squared. Multiplying the outside terms gives us negative two 𝑥. Multiplying the inside terms gives us six 𝑥. And finally, multiplying the last terms gives us negative four. We are now left with four 𝑥 squared minus 𝑥 is equal to three 𝑥 squared minus two 𝑥 plus six 𝑥 minus four.

As this is a quadratic equation, we need to get it equal to zero. Subtracting three 𝑥 squared from both sides of the equation gives us 𝑥 squared. Subtracting four 𝑥 from both sides of the equation gives us negative five 𝑥 on the left-hand side and zero 𝑥 on the right-hand side. Finally, adding four to both sides of the equation gives us plus four on the left and zero on the right. We are now left with the quadratic equation 𝑥 squared minus five 𝑥 plus four equals zero.

In order to solve this quadratic equation, we’ll factorize it into two brackets. The first term in both brackets will be 𝑥 as 𝑥 multiplied by 𝑥 gives us 𝑥 squared. The second terms in both of the brackets must multiply to give us positive four and add to give negative five. The only pair of numbers that satisfies both of these are negative four and negative one. Negative four multiplied by negative one is positive four and negative four plus negative one is equal to negative five.

This means that the second term in our brackets are negative one and negative four. As one of these brackets must be equal to zero, we end up with two possible values of 𝑥: either 𝑥 is equal to one or 𝑥 is equal to four. In order to check these answers, we can substitute 𝑥 equals one and 𝑥 equals four back into the initial ratios.

When we substitute in 𝑥 equals one, we end up with three to three is equal to one to one. As these ratios are equal, the solution 𝑥 equals one is correct. Substituting in 𝑥 equals four gives us the ratios 15 to six and 10 to four. Both of these ratios can be simplified to five to three. Therefore, the solution 𝑥 equals four is also correct.

There are two possible values of 𝑥: 𝑥 equals one and 𝑥 equals four.

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