I’m stuck on something super basic with fractions and it’s driving me a little bananas. I was making lemonade and the recipe said add 2/3 cup of sugar and then 3/5 cup more. My instinct was to just do 2/3 + 3/5 = (2+3)/(3+5) = 5/8, but that felt wrong because 2/3 is about 0.67 and 3/5 is 0.6, so the total should be more than 1 cup, not 5/8 of a cup!
Then I tried the “common denominator” thing: LCM of 3 and 5 is 15, so I rewrote them as 10/15 and 9/15. But then I got confused about what to do next. Do I add just the numerators? I wrote 10/15 + 9/15 = 19/30, but that also seems off because it’s still less than 1. My brain keeps wanting to add both top and bottom to make it feel fair, but everyone keeps telling me you only add the numerators once the denominators match. Why exactly is that?
Is there a simple picture way to see it? Like with pizza slices: if I cut one pizza into thirds and another into fifths, does finding a common denominator mean I’m re-slicing all the pieces so they’re the same size, and then I just count how many equal slices I have? If that’s the idea, why wouldn’t I add the bottoms too?
Also, how do you decide on a good common denominator quickly without making it huge? I picked 15 here, but I sometimes jump to something like 60 just because it feels “safe,” and then the numbers get messy.
Any help appreciated!
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3 Responses
You’re totally on the right track with the pizza picture: the denominator is the size of the slice, and the numerator is how many of those equal slices you have. When you “find a common denominator,” you’re just re-slicing so both piles use the same slice size; after that, you only add the counts of slices, not the slice size. Adding denominators would change the unit mid‑add, which is like saying “3 inches + 4 inches = 7 inch-inches” (eep!). In your lemonade, 2/3 = 10/15 and 3/5 = 9/15, so 10/15 + 9/15 = 19/15, not 19/30. That’s 1 4/15 ≈ 1.27 cups, which matches your gut that it should be more than 1. A quick “why not add tops and bottoms” check: 1/2 + 1/2 should be 1, but (1+1)/(2+2) = 2/4 = 1/2, so that method fails. As for picking a denominator, use the least common multiple to keep things small: list a few multiples (3: 3,6,9,12,15; 5: 5,10,15 → boom, 15), or factor (3 = 3, 5 = 5, so LCM = 3×5 = 15). If denominators share factors (like 6 and 8), combine the highest powers of each prime (LCM of 6 and 8 is 24), and you’re golden. Hope this helps!
That itchy “add the tops and the bottoms” instinct is super common, but the denominator is the size of the slice, not something you’re counting. When we add fractions, we only want to count how many equal-sized slices we have, so we keep the slice size fixed and add the counts. If you added denominators, you’d be secretly changing the size of the slices mid-addition. Quick check: 1/2 + 1/2 should be 1, but (1+1)/(2+2) = 2/4 = 1/2 (whoops). For your lemonade: 2/3 = 10/15 and 3/5 = 9/15, so 10/15 + 9/15 = 19/15, not 19/30. That’s 1 and 4/15 cups, which matches your gut that it should be more than 1. I might be over-personifying the slices, but the rule really is: once the slice size matches, only add the counts.
Picture-wise, yes: think of re-slicing both pizzas into fifteenths so every piece is the same size, then just count how many of those equal slices you’ve got. The denominator stays 15 because you didn’t change the slice size after lining them up; you just gathered them together. For choosing a good common denominator: use the least common multiple (LCM). If one denominator divides the other, use the larger (e.g., 3 and 12 → 12). If they’re coprime, multiply them (3 and 5 → 15). Otherwise factor and take the highest powers (6 and 8 → 2^3·3 = 24). A catch-all shortcut is a/b + c/d = (ad + bc)/bd, then simplify; you can often cancel crosswise before multiplying to keep numbers friendly. I’m pretty sure that covers it-and your lemonade doesn’t need a calculator, just nicely matched slices.
Think of the denominator as the slice size and the numerator as how many slices: re-cut both to fifteenths, then just count 10 + 9 = 19 equal slices, so 2/3 + 3/5 = 19/15 (>1); adding denominators would be like changing the slice size mid-count, which only really makes sense if the slices were already the same size. For speed, use the least common multiple (here 15); using 60 also “works,” it just makes extra tiny slices that reduce back to the same total.