Before buying the $45 book,
Ratio of Grace $ : Chris $ : Total $ : Difference
= 3 : 7 : 10 : 4
After buying the $45 book,
Ratio of Grace $ : Chris $ : Total $ : Difference
= 1 : 4 : 5 : 3
Now since they each bought a book of the same cost, the amount spent is the same. So the difference between their money left and difference between their money at first is the same. It doesn't change.
We have to make the differences for each ratio to become the same.
Common multiple of 4 and 3 is 12 because 4 x 3 = 12
Multiply by 3, Ratio of Grace $ : Chris $ : Total $ : Difference before buying book
= 9 : 21 : 30 : 12
Multiply by 4, Ratio of Grace $ : Chris $ : Total $ : Difference after buying book
= 4 : 16 : 20 : 12
9 - 4 = 5, and 21 - 16 = 5. So they each spent 5 units on the book.
5 units = $45 so 1 unit = $45 ÷ 5 = $9
Total amount they had at first = 30 units
= $9 x 30 = $270