Find the value of 𝑎 plus two 𝑏 plus 𝑐 given 𝑎 plus 𝑏 is equal to two over five and 𝑏 plus 𝑐 equals to four over seven.
On inspection of our original expression, we can see that we have one 𝑎 term, two 𝑏 terms, and one 𝑐 term. And then if we have a look at what we’re given, so 𝑎 plus 𝑏 is equal to two-fifths and 𝑏 plus 𝑐 is equal to four over seven, we can actually see that we’ve got again one 𝑎.
We’d actually have two 𝑏s if we added these together. And we’d have a 𝑐. So therefore, we can say that if we add them together, we get 𝑎 plus 𝑏 plus 𝑏 plus 𝑐, which is gonna be equal to our original expression of 𝑎 plus two 𝑏 plus 𝑐. So, great! We now know that if we add these together, we get the same as our original expression.
So therefore, we can actually surmise that 𝑎 plus two 𝑏 plus 𝑐 must be equal to two-fifths plus four-sevenths. Now to add these fractions together, the first thing we want to do is find a common denominator. In this case, it’ll be 35 because both five and seven go into 35.
So then we have a look and see what we had to multiply our denominator from- by to get our common denominator. So in the first fraction, five we have to multiply it by seven to get 35. And our second fraction is gonna be seven multiplied by five to get to 35.
And we’re now gonna do the same to the numerators. So we’re gonna multiply our two by seven, which is gonna give us 14. And then we’re gonna multiply our four by five, which will give us 20. So now we have a pair of fractions that we can add together because they have a common denominator.
When the fractions have a common denominator, all we have to do is add the numerators. So we’ve got 14 plus 20 over 35. So therefore, we can say that our final answer is that 𝑎 plus two 𝑏 plus 𝑐 is equal to 34 over 35. And that’s because 14 plus 20 is 34, so 34 over 35.