Evaluate seven 18ths plus four 15ths plus one-ninth giving the answer in its simplest form.
In order to add fractions, they must always have a common denominator. And that means we should try and find the least common multiple of 18, 15, and nine. To do that, let’s consider the multiples of nine, 15, and 18. Nine times two is 18. Nine times three is 27. Nine times four 36. Nine times five 45. Nine times six 54. Nine times seven 63. Nine times eight 72. Nine times nine 81. And nine times 10 is 90.
We’ll do the same thing for 15. 15 times two equals 30. 15 times three equals 45. We see that nine and 15 have a common multiple at 45. Now we need to check and see if that’s a multiple of 18. 18 times two is 36. That means nine and 18 are both a multiple of six. But we need a multiple that all three of them share. So, we’ll have to keep going.
18 times three equals 54. That’s a match with nine but not with 15. 18 times four equals 72. This pattern is here because every multiple of 18 is also a multiple of nine. In fact, 18 is a multiple of nine. So, let’s go back to 15 and focus on that one. 15 times four is 60. 15 times five is 75. 15 times six is 90. And we have a match for nine and 15.
Going back to 18, if we multiply 18 by five, we get 90. The least common multiple for nine, 15, and 18 is 90. Nine times 10 equals 90. 15 times six equals 90. And 18 times five equals 90.
We now wanna take all three of our fractions and turn them into values with denominators of 90. 18 times five equals 90. And if we multiply the denominator by five, then we multiply the numerator by five. Seven times five equals 35 And that means seven 18ths is the same thing as 35 93ths. Doing the same thing to four 15ths, 15 times six equals 90. And four times six equals 24. One-ninth, nine times 10 equals 90. And one times 10 equals 10.
At this stage, we can check that, yes, we do have a common denominator. And once we have a common denominator, we can add all the numerators together. 35 plus 24 plus 10 equals 69. And the denominator doesn’t change; it stays 90. The three of these fractions added together equals 69 90ths.
But we need to simplify. We need to give our answer in the simplest form. And that means we need to know is there anything we can divide the numerator and the denominator by to simplify this fraction. Both 69 and 90 are divisible by three. 69 divided by three equals 23. And 90 divided by three equals 30. We’ll check again. Is there anything that 23 and 30 are both divisible by? Well, 23 is a prime number. And that means it’s only divisible by one and itself. This confirms that this fraction is in its simplest form. And our final answer is 23 30ths.