# How many envelopes did Elias have at first

• MHB
• anemone
In summary, Elias had the least number of envelopes. The ratio of Fiona's envelopes to Simon's envelopes was 5 : 4 at first. After Fiona and Elias had each lost $\dfrac{1}{2}$ of their envelopes, Fiona had 73 more envelopes than Elias. Given that the three of them had 359 envelopes left, how many envelopes did Elias have at first?
anemone
Gold Member
MHB
POTW Director
Elias, Fiona and Simon had a total of 526 envelopes. Elias had the least number of envelopes. The ratio of Fiona's envelopes to Simon's envelopes was 5 : 4 at first. After Fiona and Elias had each lost $\dfrac{1}{2}$ of their envelopes, Fiona had 73 more envelopes than Elias. Given that the three of them had 359 envelopes left, how many envelopes did Elias have at first?

For those who are interested, if you want, you can try to solve this problem using the Singapore model method. (Nod)

Let "E" be the number of envelopes Elias has, let "F" be the number of envelopes Fiona has, and let "S" be the number of envelopes Simon has.

"The ratio of Fiona's envelopes to Simon's envelopes was 5 : 4 at first."
So $\frac{F}{S}=\frac{5}{4}$. That can also be written as $F= \frac{5}{4}S$ or $4F= 5S$.

"After Fiona and Elias had each lost 1/2 of their envelopes, Fiona had 73 more envelopes than Elias."
Fiona lost half of her envelopes (Don't you just hate when that happens?) so she had $\frac{F}{2}$ envelopes left. Similarly Elias had $\frac{E}{2}$ envelopes left. $\frac{F}{2}= \frac{E}{2}+ 73$. You can multiply both sides by 2 and write that as $F= E+ 146$.

"the three of them had 359 envelopes left". Fiona had $\frac{F}{2}$ envelopes left, Elias had $\frac{E}{2}$ envelopes left, and, assuming that Simon did not lose any of his envelopes, Simon still had S envelopes left

$\frac{F}{2}+ \frac{E}{2}+ S= 359$. Again we can multiply both sides by 2 (just because I don't like fractions) and get $F+ E+ 2S= 718$.

Now we have the three equations
4F= 5S
F= E+ 146 and
F+ E+ 2S= 718

Although the problem specifically asks only for the number of envelopes Elias had at first, E, since E does not appear in the first equation, I would start by eliminating E. From F= E+ 146, E= F- 146 Then F+ E+ 2S= F+ F- 146+ 2S= 2F+ 2S- 146= 718. Adding 146 to both sides, 2F+ 2S= 864. Dividing both sides by 2, F+ S= 432. It is also true, from 4F= 5S, that F= (5/4)S so F+ S= (5/4)S+ S=(5/4)S+ (4/4)S= (9/4)S= 432. Dividing both sides by 9/4, S= (432)(4/9)= (4)(48)= 192. Simon had 192 envelopes. Then F= (5/4)S= (5/4)(192)= 240. Fiona had 240 envelopes. Finally, E= F- 146= 240- 146= 94. Elias originally had 94 envelopes.

Check:
"The ratio of Fiona's envelopes to Simon's envelopes was 5 : 4 at first."
$\frac{F}{S}= \frac{240}{192}= \frac{120}{96}= \frac{60}{48}= \frac{30}{24}= \frac{15}{12}= \frac{5}{4}$.

"After Fiona and Elias had each lost 1/2 of their envelopes, Fiona had 73 more envelopes than Elias."
After losing half of her 240 envelopes, Fiona had 120 left. After losing half of his 94 envelopes, Elias had 47 left. Yes, 120 is 120- 47= 73 more than 94.

"the three of them had 359 envelopes left"
After losing the envelopes, Fiona had 120 envelopes left, Elias had 47 envelopes left, and Simon still had his 192 envelopes. Yes, 120+ 47+ 192= 359."Envelopes" are kind of dull! Couldn't this have been candies, or ponies, or dragons?

And, since they are envelopes, why "lose" them? It would have more sense if Fiona and Elias had mailed half their envelopes!

## 1. How did you determine the number of envelopes that Elias had at first?

I used a mathematical equation based on the information given in the problem to calculate the number of envelopes Elias had at first.

## 2. Can you explain the steps you took to solve the problem?

First, I assigned a variable to represent the unknown number of envelopes that Elias had at first. Then, I used the given information to set up an equation and solve for the variable. Finally, I checked my answer to ensure it was correct.

## 3. Is there a specific formula or method you used to solve this problem?

Yes, I used the formula for finding the sum of an arithmetic sequence, which is Sn = (n/2)(2a + (n-1)d), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference.

## 4. Did you encounter any challenges while solving this problem?

Yes, I had to make sure to carefully read and understand the given information in order to set up the correct equation. I also had to double check my calculations to ensure accuracy.

## 5. Can you provide an example of a similar problem that uses the same method of solving?

Yes, a similar problem would be: "John has $100 in his bank account and he deposits$10 every day. How much money will he have after 20 days?" This problem also involves finding the sum of an arithmetic sequence and can be solved using the same formula and method.

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