Who ate the most sweets
Sophie, Ben, and Tara each have a bag of sweets.
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Sophie eats half of her sweets and gives away 6.
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Ben eats a third of his sweets and gives away 12.
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Tara eats a quarter of her sweets and gives away 18.
They each end up with the same number of sweets left.
Who started with the most sweets?
Let:
ReplyDeleteS be the number of sweets Sophie started with
B be the number of sweets Ben started with
T be the number of sweets Tara started with
Sophie:
Eats half: ½S
Gives away 6
Left with: S - ½S - 6 = ½S - 6
Ben:
Eats a third: ⅓B
Gives away 12
Left with: B - ⅓B - 12 = ⅔B - 12
Tara:
Eats a quarter: ¼T
Gives away 18
Left with: T - ¼T - 18 = ¾T - 18
All three are left with the same number of sweets:
So,
½S - 6 = ⅔B - 12 = ¾T - 18
Let’s solve step-by-step.
Step 1: Set Sophie and Ben’s sweets equal
½S - 6 = ⅔B - 12
Multiply both sides by 6 to eliminate fractions:
6(½S - 6) = 6(⅔B - 12)
3S - 36 = 4B - 72
Now solve:
3S - 4B = -72 + 36
3S - 4B = -36 → Equation (1)
Step 2: Set Ben and Tara’s sweets equal
⅔B - 12 = ¾T - 18
Multiply both sides by 12 to eliminate fractions:
12(⅔B - 12) = 12(¾T - 18)
8B - 144 = 9T - 216
Now solve:
8B - 9T = -216 + 144
8B - 9T = -72 → Equation (2)
Step 3: Solve the system of equations
From Equation (1):
3S = 4B - 36 → So, S = (4B - 36)/3
From Equation (2):
8B - 9T = -72 → Solve for B:
8B = 9T - 72
So, B = (9T - 72)/8
Now substitute this into the expression for S:
S = (4 * [(9T - 72)/8] - 36) / 3
= ( (36T - 288)/8 - 36 ) / 3
= ( (36T - 288 - 288)/8 ) / 3
= (36T - 576)/8 ÷ 3
= (36T - 576)/(8 × 3)
= (36T - 576)/24
So we now know:
S = (36T - 576)/24
B = (9T - 72)/8
Try a value for T to find whole numbers for S and B.
Try T = 48:
B = (9×48 - 72)/8 = (432 - 72)/8 = 360/8 = 45
S = (36×48 - 576)/24 = (1728 - 576)/24 = 1152/24 = 48
Now check how many sweets each has left:
Sophie: Started with 48
Eats half: 24
Gives away 6
Left: 18
Ben: Started with 45
Eats a third: 15
Gives away 12
Left: 18
Tara: Started with 48
Eats a quarter: 12
Gives away 18
Left: 18
✅ They all have 18 left.