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Ms. Leong owned an ice cream store and had chocolate and vanilla ice cream for sale. She sold [tex]\frac{3}{4}[/tex] of the chocolate ice cream and [tex]\frac{1}{2}[/tex] of the vanilla ice cream. If she had 20 scoops of chocolate ice cream and 30 scoops of vanilla ice cream left, how many scoops of ice cream did Ms. Leong have altogether at first?

Sagot :

To determine the initial amount of ice cream Ms. Leong had, we need to work backwards from the amount left after sales.

1. Chocolate Ice Cream:
- Ms. Leong sold [tex]\(\frac{3}{4}\)[/tex] of her chocolate ice cream.
- After selling, she had 20 scoops left.

Let's denote the initial amount of chocolate ice cream as [tex]\(C\)[/tex]. Since she sold [tex]\(\frac{3}{4}\)[/tex] of it, the remaining [tex]\(\frac{1}{4}\)[/tex] is what she has left.

Therefore, we can write the equation:
[tex]\[ \frac{1}{4}C = 20 \][/tex]

Solving for [tex]\(C\)[/tex], we multiply both sides by 4:
[tex]\[ C = 4 \times 20 = 80 \][/tex]

So, initially, she had 80 scoops of chocolate ice cream.

2. Vanilla Ice Cream:
- Ms. Leong sold [tex]\(\frac{1}{2}\)[/tex] of her vanilla ice cream.
- After selling, she had 30 scoops left.

Let's denote the initial amount of vanilla ice cream as [tex]\(V\)[/tex]. Since she sold [tex]\(\frac{1}{2}\)[/tex] of it, the remaining [tex]\(\frac{1}{2}\)[/tex] is what she has left.

Therefore, we can write the equation:
[tex]\[ \frac{1}{2}V = 30 \][/tex]

Solving for [tex]\(V\)[/tex], we multiply both sides by 2:
[tex]\[ V = 2 \times 30 = 60 \][/tex]

So, initially, she had 60 scoops of vanilla ice cream.

3. Total Initial Amount of Ice Cream:
- Now, we just add the initial amounts of chocolate and vanilla ice cream to find the total.
[tex]\[ \text{Total} = C + V = 80 + 60 = 140 \][/tex]

Therefore, Ms. Leong initially had 140 scoops of ice cream.