# Calculate Total based on Percent Change

I'm trying to help my kid answer this math question. Every example I've found on the internet is far too simple. Can someone help me answer this? And, what type of question is this? Thanks!

Jaden and Leah are wrapping presents. Jaden has wrapped 20% more presents than Leah. Then Leah wraps 10 more presents. Now, Leah has wrapped 40% more presents than Jaden. How many presents has Jaden wrapped?

• My logic is J = 1.2L and L + 10 = 1.4J but the result is not nice. Feb 8 '20 at 1:01
• @SSpring $25$ is almost $38.9\%$ more than $18$, which is close to $40\%$ even if not equal to it Feb 8 '20 at 1:04

This is a word problem (okay - well, duh). You might see it called a ratios problem.

As with all word problems, the trick is to assign variables to the unknowns. The obvious ones here are:

• Let $$J$$ be the number of presents that Jaden wrapped.
• Let $$L$$ be the number of presents that Leah wrapped at first.

Then we are told that Jaden wrapped 20% more presents than Leah. This is where the wording gets tricky. It means that the difference between the number of presents that Jaden and Leah wrapped is 20% of Leah's total. So $$J - L = \frac {20}{100} L = \frac 15 L$$ We can multiply through by $$5$$ $$5J - 5L = L$$ and add $$5L$$ to both sides $$5J = 6L$$ or $$L = \frac 56 J$$

Now we are also told that Leah wraps 10 more presents. So her number is now $$L + 10$$, while Jaden apparently took a break. Now she has wrapped 40% more presents than he did. So, $$(L + 10) - J = \frac{40}{100}J = \frac 25J$$ Multiplying through again $$5L + 50 - 5J = 2J$$ $$5L + 50 = 7J$$ Substitute in $$L = \frac 56J$$ to get

$$5\left(\frac 56J\right) + 50 = 7J$$ multiply through by the fraction $$25J + 300 = 42J$$ combine the terms in $$J$$ $$300 = (42 - 25)J = 17J$$ and divide by the coefficient $$J = \frac {300}{17} = 17\frac {11}{17}$$

So Jaden wrapped $$17$$ and $$\frac{11}{17}$$ presents, Leah (in the end) wrapped $$24$$ and $$\frac{12}{17}$$ presents, leaving unsettling questions about their work ethic and how one measures the partial wrapping of presents.

Okay - I really shouldn't leave it at that flippant remark. The above "solution" is the only possible way to get exactly 20% and 40%. But since fractional wrappings are not possible, obviously they each wrapped an integer number of presents. So we can conclude that the 20% and/or 40% are approximate values, not exact. Since both fractions are greater than $$\frac 12$$, we can try rounding them up: Jaden wrapped $$18$$ presents, while Leah wrapped $$15$$ initially, and $$25$$ total.

$$\frac{18 - 15}{15} = 0.20 = 20\%\\\frac{25 - 18}{18} \approx 0.39 = 39\%$$

so it appears that the author of the problem did the dastardly deed of rounding 39% up to 40%