How many envelopes did Elias have at first

[anemone]

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)

 

[HOI]

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!