If seventeen elevenths of the sum of two numbers is 51, where the ratio between them is four to seven, find the two numbers.
The first thing we need to do in this question is to actually find the sum of these two numbers. We’re told that seventeen elevenths of the sum of these two numbers is 51. And since the numerator, 17, is larger than the denominator, this means that the actual sum will be less than 51. Let’s define 𝑥 to be equal to the sum of the two numbers.
Since seventeen elevenths of the sum of the two numbers is 51, we can write this mathematically as seventeen elevenths 𝑥 equals 51. In order to work out the value of 𝑥, we rearrange the equation. Multiplying at both sides of our equation by 11 gives us 17𝑥 equals 51 times 11. We then evaluate the right-hand side to give us the equation 17𝑥 equals 561. To find out 𝑥 then, we divide both sides of our equation by 17. So, 𝑥 equals 33.
And now we know that the sum of the two numbers is 33. We now split the sum of the numbers into the ratio of parts, so that number one has four parts and number two has seven parts. To find the total parts that we split our value 33 into, then we add our parts four and seven to give us 11. We then divide the sum of our numbers, 33, by the total parts, which is 11, to give three. So, the value of each part is equal to three.
So, to find the value of our first number, we take the four parts and multiply it by the value three, giving us the answer 12 for number one. In the same way, to find our second number, we take our seven parts and multiply it by the value three. And so, our second number is 21.
Therefore, our final answer is the two numbers are 12 and 21.