Initially, Alice had 28 sweets and Olivia 12 sweets.
Step-by-step explanation:
First assume that Aunty Joan has x number sweets.
Alice and Olivia shared the x sweets in a ration of 7:3, this means
Alice had 7/10 x sweets and Olivia had 3/10 sweets.
Alice gives 3 sweets to Olivia, Alice will remain with ;
[tex]\frac{7}{10} x-3[/tex]
Then the new ratio changes to 5:3 which means Alice will have 5/8 x number of sweets.
Equate the number of sweets for Alice in the two cases, which is
[tex]\frac{5}{8}x=\frac{7}{10} x-3\\ \\\frac{7}{10} x-\frac{5}{8} x=3\\[/tex]
[tex]\frac{56x-50x}{80} =3\\\\\\\frac{6}{80} x=3\\\\\\6x=80*3\\\\\\x=240/6\\\\x=40[/tex]
So there were 40 sweets at first
Using the ratio of 7:3 then
Alice had 7/10 *40 = 28 sweets
Olivia had 3/10*40= 12 sweets
Alice gave Olivia 3 sweets, so she will remain with 28-3 =25 sweets. Olivia will now have 15 sweets.The new ratio will be 25:15 simplified as 5:3.
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