If the ratio of a triangle’s side lengths is one-quarter to one-sixth to one-third and its perimeter is 23, find the respective lengths of each side of the triangle.
Well, the first thing we need to do is make our ratio equivalent because we’ve got one-quarter to one-sixth to one-third. So, what we want to do is have each of our fractions with the same denominator. So, we’re gonna have to find a common denominator. Well, if we start to list the multiples of four, six and three, we can see that the least common multiple is going to be 12. And that’s because four goes into 12 three times, six goes into 12 twice, and three goes into 12 four times.
So therefore, if we convert each of our fractions to twelfths, the first fraction is gonna be three twelfths. And that’s because if we multiply four by three to get to 12, then we have to do the same to the numerator. So, one multiplied by three is three. Then, we’ve got two twelfths for a sixth and four twelfths for a third. So now, what we can do is concentrate on our numerators. And that’s because our denominators are the same. So, we’ve got three twelfths to two twelfths to four twelfths.
So therefore, if we multiply through, which we can do, our ratio by 12, this will give us the ratio three to two to four. Okay, great, so now we’ve got this. What we can do is we can use this to find the respective length of each side of the triangle. So first of all, what we want to do is find the total number of parts in our new ratio and to do that we add three, two, and four. And when we do that, we get nine as the answer. So, we know there are nine parts in total.
We were also told that the sum of the side, because the perimeter of the triangle is the sum of the lengths of all the sides, is equal to 23. So therefore, we can say that nine parts is equal to 23. So, what we want to do now is find out what one part is. Well, to find one part, we divide both sides by nine. And that’s because nine parts divided by nine is one part. So, we get one part is equal to 23 over nine. So now, what we want to do is work out each section of our ratio.
So first of all, we’ve got three parts. So, we do three multiplied by 23 over nine, which is the value of one part. And this gives us 69 over nine. And that’s because if you have three multiplied by 23, it’s 69. And three can be thought of as three over one. And when you multiply fractions, you multiply numerators and denominators.
And if we convert this into a mixed number, it’s seven and six-ninths. That’s because nine goes into 69 seven times with the remainder of six. And then, we can simplify six-ninths cause we can divide the numerators and denominators by three. So therefore, this gives us our first length of seven and two-thirds.
Now, we’ll move on to the next length. So, for our next length, we have two multiplied by 23 over nine. Well, this gives us 46 over nine. Well, if we divide 46 by nine, we get five remainder one. So therefore, this gives us five and one-ninth. So, the next length is five and one-ninth.
So now, we can move on to the final length. Well, for the final length, we have four multiplied by 23 over nine. And this gives us 92 over nine. Well, if we divide 92 by nine, we get 10 remainder two. So, we get 10 and two-ninths. So therefore, we can say that the correct lengths for each side of the triangle are seven and two-thirds, five and one-ninth, and 10 and two-ninths.
Well, what we can do to double-check the answer is add together our lengths to make sure they’re equal to 23. So, we’ve got seven and two-thirds plus five and one-ninth plus 10 and two-ninths. Well, if we add together the units first, we get seven plus five plus 10, which is equal to 22.
And then, we’re gonna have plus six over nine. And that’s because if we convert two-thirds to ninths, we multiply the numerator and denominator by three. Then, plus one over nine plus two over nine. And this gives us nine over nine. So therefore, we’ve got 22 plus one which equals 23, which is what we were looking for. So therefore, yes, definitely we’ve checked the answer and it is correct.