The table shows the weights of 64 students. Find the value of 𝑘.
So if we take a look at the table, what we can see is that we’ve got 𝑘 plus five students between 50 and 54 kilograms, three 𝑘 between 55 and 59. 60 to 64 is two 𝑘. Then we’ve got two 𝑘 plus four from 65 to 69 kilograms, three 𝑘 minus 16 from 70 to 74 kilograms, and finally 𝑘 minus one from 75 kilos plus.
So therefore, to solve this problem, we can say that the sum of each of our number of students is gonna be equal to 64. But why is it equal to 64? Well, it’s equal to 64 because we’re told that the table shows the weights of 64 students in the question. And also we have a total column in the table which has 64.
So now the equation we have is 𝑘 plus five plus three 𝑘 plus two 𝑘 plus two 𝑘 plus four plus three 𝑘 minus 16 plus 𝑘 minus one is equal to 64. So now what we need to do is collect like terms. So I’m gonna begin with the 𝑘-terms. And as you’ve seen, what we’ve done is circled them. But I’ve also circled the sign before them because we know that this sign belongs to them. So it tells us if it’s positive or negative.
So we have 𝑘 plus three 𝑘, which is four 𝑘, plus two 𝑘, which takes us to six 𝑘, plus another two 𝑘, which takes us to eight 𝑘, plus three 𝑘, which takes us to 11𝑘, plus one more 𝑘, which takes us to 12𝑘. So now I’m gonna collect the numerical terms. So we have positive five plus four, which is nine, minus 16, which is gonna be equal to negative seven, and then minus another one, which gonna be negative eight. So therefore, the equation we now have is 12𝑘 minus eight is equal to 64.
So what we want to do now is to solve this to find the value of 𝑘. The first step in solving this will be adding eight to each side of the equation. And when I do that, I’m gonna get 12𝑘 — and that’s because if you have 12𝑘 minus eight and then add eight to it, it’s just gonna leave us with 12𝑘 because negative eight plus eight is zero — is equal to 72.
So now what we need to do is divide each side of the equation by 12. And when we do that, we get 𝑘 is equal to six. And that’s because 12𝑘 divided by 12 just gives us 𝑘. And 72 divided by 12 is six. So therefore, we can say that the value of 𝑘 is going to be equal to six.
And what we can do now is check if this works. And the way we can do this is by substituting in 𝑘 is equal to six into each of our columns in the table. So we do that with the first column. We’re gonna have six plus five, which is 11. Then the second column will be three multiplied by six, which is 18. The third column will be two multiplied by six, which is 12. The fourth column will be two multiplied by six, which is 12, plus four, which is 16. Then the fifth column is gonna be three multiplied by six, which is 18. 18 minus 16 is two. And then, finally, our sixth and final column, we’ve got 𝑘 minus one, so six minus one, which is five.
So now to check our answer, what we can do is add these all up. So first of all, we start with the units. So we’ve got one plus eight, which is nine, add two is 11, add six is 17, add two is 19, and add five is 24. So we put a four in the units column and carry the two into the tens column. So then we’ve got one add one, which is two, add another one is three, add another one is four, add the two that we carried is six. So therefore, when we’ve added them together, we get 64, which is right because that was the total number of students. So therefore, we’ve double checked and we can confirm that 𝑘 is equal to six.