Sage L. answered • 03/30/21
An Academic Guide That Inspires Confidence
To find how much pizza everyone ate, we need to add the amount of pizza everyone ate. So, we need to add 2/3, 7/12, 5/12, and 5/6. In order to add all these fractions with different denominators (different numbers in the bottom part of the fraction), we need to find a the least common denominator (LCD). The LCD is the lowest number that is a common multiple shared between all the denominators you want to add (or subtract). For context, a multiple is the product of a number times another number. For example, 4 is a multiple of 2 because 2 x 2 = 4. 6 is also a multiple of 2 because 2 x 3 = 6. 2 is also a multiple of 2 because 2 x 1 = 2. For this problem, we will only be considering multiples that are a result from multiplying a number by an integer (non-fraction or non-decimal value like 1, 2, 3, 4 and not numbers like 1/2 or 0.5).
We need to find a number that is a multiple of all the denominators of the four fractions. Let's look at the multiples of 3 from the denominator of 2/3. The first multiple of 3 we can look at is 3 because 3 x 1 = 3. But 3 is not a multiple of 12 or 6 (these are the denominators of 7/12, 5/12, and 5/6). 12 multiplied by an integer can never be 3, and 6 multiplied by any integer can never be 3 either. This means that 3 is not the LCD. Let’s try 6 next because 3 x 2 = 6. 6 is a multiple of 6, but it is not a multiple of 12. This means that 6 is not the LCD. Let’s try 9 next because 3 x 3 = 9. 9 is not a multiple of 6 or 12, so 9 is not the LCD. Let’s try 12 because 3 x 4 = 12. 12 is a multiple of both 6 and 12 because 6 x 2 = 12 and 12 x 1 = 12! Since 12 is a multiple of all the denominators, 12 is our LCD!
Now that we have now our LCD, we must convert our fractions so that our denominators equal 12. Let’s look at 2/3. How do we convert 2/3 so that the denominator is 12? Well, we need to multiply the denominator by 4 because 3 x 4 = 12, but we also have to multiply 2 (the numerator) by 4, too. Why? Let’s say we just multiplied the denominator by 4 and not the numerator. We would have 2/12. We can simplify this further because 12 divided by 2 is 6. This means that 2/12 = 1/6 which is not equal to our original fraction 2/3. We can’t add fractions that are not equivalent to our original fractions because we would be adding completely different fractions! Here’s a rule of thumb for all fractions. Whatever you multiply the denominator by, you must also multiply the numerator by that same number. In this case, we need to multiply 2/3 by 4/4, so we are multiplying both the numerator and the denominator by the same number. Notice that 4/4 simplifies to 1/1 = 1. Multiplying anything by 1 is itself, so there have been no changes to our fraction. They are the same! 2/3 x 4/4 = 8/12. Use the same process for the other fractions, and remember that we have to multiply the numerator and denominator by the number that makes the denominator equal to 12. So, 7/12 x 1/1 = 7/12. 5/12 x 1/1 = 5/12. 5/6 x 2/2 = 10/12.
Now we can finally add the converted fractions together! You only add the numerators together, not the denominators. Starting with 8/12 + 7/12 = 15/12. Then add 5/12 to 15/12, so 15/12 + 5/12 = 20/12. Finally, 20/12 + 10/12 = 30/12. So, Milo, Amy, Chris, and Elizabeth ate a combined total of 30/12 pizzas. If you want a complex fraction which is made up of a whole number and a fraction, find out how many times the denominator goes into the numerator without remainders. 12 goes into 30 2 full times plus a remainder of 6 (12 x 2 = 24 and 30 – 24 = 6). The “full times” the denominator goes into the numerator is your whole number, and the remainder will be the numerator of the new fraction. The denominator will stay the same. So, everyone ate a combined total of 2 6/12 pizzas or 2 ½ pizzas when simplified.
How much pizza is left? You started with 4 whole pizzas. 4 whole pizzas is the same as 4/1 pizzas. 4/1 x 12/12 = 48/12 pizzas. We multiply by 12/12 so that the denominator is 12 which is the LCD we found earlier. Remember how everyone ate 30/12 pizzas? We need to subtract 30/12 from 48/12 to determine how much pizza is left after everyone ate their share. So, 48/12 – 30/12 = 18/12. 18/12 = 3/2 after simplifying by dividing the numerator and denominator by 6. 3/2 = 1 ½ pizzas after doing the complex fraction method we used earlier to convert 30/12. So, there is 3/2 pizzas or 1 ½ pizzas left (both answers are correct)!
Please let me know if you have any questions!
