Let's add some rational numbers. And I'm using that word because that's the word that this book uses, but in more popular terminology we'll be adding fractions. So let's just go through all of these, actually, just to see all of the examples. So first we're going to have 3/7 plus 2/7. Our denominators are the same, so we can just add the numerators. So our denominator is 7, 3 plus 2 is 5. That is a. Let me do every other. It would take forever to do all of them. Not forever, but just more time than I want to spend. So c is 5/16 plus 5/12. Our denominators are not the same. We have to find a common denominator, which has to be the least common-- it actually could be any common multiple of these, but for simplicity let's do the least common multiple. So what's the smallest number that's a multiple of both 16 and 12? So let's see, 16 times 2 is 32, not there yet. Times 3, 48. That seems to work. 12 goes into 48 four times. So let's use 48 as our common denominator. So we had to multiply 16 times 3 to get to 48, so we're going to have to multiply this 5 times 3. We're just multiplying the numerator and the denominator by the same number, so we're not really changing it. So 5 times 3 is 15. And then to get from this 12 to this 48 right there, we had to multiply times 4. So then to get to 5 to this numerator over here, we have to multiply times 4. 5 times 4 is 20. Now we have the same denominator. So this is going to be equal to, our denominator is 48. And so we can add 15 plus 20, which is 35. And can we reduce this? Let's see, 5 does not go into 48. 7 does not go into 48. It looks like we're all done. Let's do e right there. 8/25 plus 7 over 10. Once again, we don't have a common denominator. But we can solve that. Let's make, let's see, 50 is the smallest number that both of these go into. 25 times 2, so that's 50. 8 over 25, to go to 50 we multiply by 2. So the 8, we're going to have to multiply by 2. So it's going to be 16 over 50. And then the 7 over 10, we're going to want to put it over 50. We multiply the 10 times 5, so we have to multiply the 7 times 5. So it's going to be 35 over 50. Now that our denominators are the same, we have it over 50. 16 plus 35, what is that? 10 plus 35 is 45, plus 6 is 51. So it is 51 over 50. Problem g. Let me do it in a new color. Problem g. So here we have 7 over 15-- I'll write the second one in a different color-- plus 2 over 9. Once again, the denominators are different. Find a common denominator. What is the smallest number that both 15 and 9 go into? Let's see, 15 times 2 is 30. Nope, not divisible by 9. 15 times 3 is 45, that works. 45 is divisible by 9. So we use 45. 15 times 3 is 45, so 7 times 3 is 21. These two fractions are equivalent. Plus we're going over 45. To get from 9 to 45, we have to multiply times 5. So to get our numerator over here, we have to multiply it times 5. So 2 times 5 is 10. 2/9 is the same thing as 10/45. So now we can add. We're adding fractions of 45. 21 plus 10 is 31, and we are done. Let's do one more problem down here, a word problem. Nadia, Peter and Ian are pooling their money to buy a gallon of ice cream. Nadia's the oldest and gets the greatest allowance. She contributes 1/2 the cost. So Nadia is contributing 1/2 the cost. So that is Nadia right there. Ian is next oldest and contributes 1/3 of the cost. So Ian contributes 1/3. That is Ian. Peter, the youngest, gets the smallest allowance and contributes 1/4 of the cost. So Peter gives 1/4 of the cost. Peter contributes 1/4 of cost. They figure that this will be enough money. When they get to the checkout, they realize that they forgot about sales tax and worry there will not be enough money. Amazingly, they have exactly the right amount of money. What fraction of the cost of ice cream was added as tax? Well, let's see, if we add 1/2 plus 1/3, plus 1/4 of the cost, let's see what we get. So we have to find a common denominator, some number that is the least common multiple of 2, 3, and 4. And let's see, 4, it would have to be 12, right? 12 is divisible by 2, it's divisible by 3, and it's divisible by 4. So 1/2 is the same thing as 6/12. 2 times 6 is 12. 1 times 6 is 6. These are equivalent. 6 is 1/2 of 12. 1/3, if we use 12 as a common denominator, to go from 3 to 12 you have to multiply by 4. So you take that 4 and you multiply it by 1. 4/12 is the same thing as 1/3. And then 1/4, if you use your denominator 12, to go from 4 to 12 you have to multiply by 3, so multiply the numerator by 3 as well, you get 3. So let's add these. So 6/12 plus 4/12, plus 3/12 is going to be equal to-- our denominator's going to be 12-- it's going to be 6 plus 4, plus 3, which is equal to 6 plus 4 is 10, plus 3 is 13. So it's going to be equal to 13/12. And this is as an improper fraction. Or we could say that this is the same thing, this is equal to 12/12 plus 1/12, or we could say the same thing as 12/12 is just 1, right? 12 divided by 12 is 1. So this is 1 and 1/12. So when they pool their money, they get 1 and 1/12 of the price of the ice cream that they want to buy. So they say what fraction of the cost of ice cream was added as tax? This is the exact amount that they needed to pay. So clearly, 1 is the non-taxed price of the ice cream, so this 1/12 was the amount added as tax. So the answer to the question is 1/12 of the price was added as tax.