Question Video: Changing the subject of a Formula Mathematics

The variables 𝑡 and 𝑚 are related by the formula 𝑡 = 𝑚/(𝑚 + 1). Make 𝑚 the subject.

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Video Transcript

The variables 𝑡 and 𝑚 are related by the formula 𝑡 is equal to 𝑚 divided by 𝑚 plus one. Make 𝑚 the subject.

In order to rearrange or change the subject of a formula, we need to use our knowledge of inverse operations and the balancing method. In this question, we want to rearrange the formula so that 𝑚 is equal to an expression not containing 𝑚. As multiplication is the opposite of division, we begin by multiplying both sides of the equation by 𝑚 plus one. On the left-hand side, we have 𝑚 plus one multiplied by 𝑡, which can also be written as 𝑡 multiplied by 𝑚 plus one.

On the right-hand side, the 𝑚 plus ones cancel. So we are left with 𝑡 multiplied by 𝑚 plus one is equal to 𝑚. Distributing the parentheses, otherwise known as expanding the brackets, means we need to multiply 𝑡 by 𝑚 and 𝑡 by one. This gives us 𝑡𝑚 plus 𝑡. Therefore, 𝑡𝑚 plus 𝑡 is equal to 𝑚.

Our next step is to get all our terms with 𝑚 in on one side of the equals sign. We can do this by subtracting 𝑡𝑚 from both sides. We are now left with 𝑡 is equal to 𝑚 minus 𝑡𝑚. We notice that 𝑚 is common in both terms on the right-hand side. Therefore, we can factor or factorize this out. 𝑡 is therefore equal to 𝑚 multiplied by one minus 𝑡.

Our final step is to divide both sides of this equation by one minus 𝑡. We leave the left-hand side as 𝑡 divided by one minus 𝑡. Canceling the one minus 𝑡 on the numerator and denominator of the right-hand side gives us 𝑚. 𝑚 is therefore equal to 𝑡 divided by one minus 𝑡. This could also be written as negative 𝑡 divided by 𝑡 minus one. Fractions are equivalent if we multiply the numerator and denominator by the same value.

In this case, we have multiplied the numerator and denominator by negative one to switch between our two answers. 𝑚 is equal to 𝑡 divided by one minus 𝑡 or negative 𝑡 divided by 𝑡 minus one.

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