Jill, Adina, and Susan had $837 all together. Jill had the least amount of money. Before they went to the store, the ratio of Adina’s money to Susan’s money was 4:3. Then Jill and Adina each spent a third of their money. They counted their money again and had a total of $648 left. How much money did Jill have at first?

Here’s my solution – let me know if you or your students solved it differently:

The girls went from $837 in total to $648 in total. So they spent 837 – 648 = $189.

Where did that $189 come from? We are told that Jill and Adina each spent a third of their money. Using the Distributive Property, we know that

1/3 x Adina’s money + 1/3 x Jill’s money = 1/3 x (Adina’s money + Jill’s money)

so $189 is one third of the sum they had at first. That means they had three times as much as $189 to start, or $567.

If the three girls had $837, and $567 of that was the sum of Adina and Jill’s money, then Susan had 837 – 567 = $270.

Now we can use the ratio relationship that we were given to find out how much Adina had at first. We know that the ratio of Adina’s money to Susan’s money was 4:3. Susan had $270. If we divide $270 into 3 equal parts, each part is $90. Adina had 4 of these equal parts, so she had $360.

Now remember that the sum of Jill’s money and Adina’s money was $567, so Jill had 567 – 360 = $207.

Answer: Jill had $207 at first.