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Dalia has 3030 more stickers than Huda at first. Dalia gave 40%40\% of her stickers to Huda. Then, Huda gave 50%50\% of her stickers to Dalia. In the end, Dalia has 4848 more stickers than Huda. How many stickers did Dalia have at first?

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Full solution

Q. Dalia has 3030 more stickers than Huda at first. Dalia gave 40%40\% of her stickers to Huda. Then, Huda gave 50%50\% of her stickers to Dalia. In the end, Dalia has 4848 more stickers than Huda. How many stickers did Dalia have at first?
  1. Define Variables: Let DD be the number of stickers Dalia had at first and HH be the number of stickers Huda had at first. We know D=H+30D = H + 30.
  2. Calculate Dalia's Remaining Stickers: Dalia gave 40%40\% of her stickers to Huda, so she gave away 0.4D0.4D stickers. After giving, Dalia has D−0.4D=0.6DD - 0.4D = 0.6D stickers left.
  3. Calculate Huda's Stickers After Receiving: After receiving stickers from Dalia, Huda has H+0.4DH + 0.4D stickers.
  4. Calculate Huda's Remaining Stickers: Then Huda gave 50%50\% of her stickers to Dalia, which is 0.5(H+0.4D)0.5(H + 0.4D). After giving, Huda has H+0.4D−0.5(H+0.4D)=0.5H+0.2DH + 0.4D - 0.5(H + 0.4D) = 0.5H + 0.2D left.
  5. Calculate Dalia's Total Stickers: Dalia now has 0.6D+0.5(H+0.4D)=0.6D+0.5H+0.2D=0.8D+0.5H0.6D + 0.5(H + 0.4D) = 0.6D + 0.5H + 0.2D = 0.8D + 0.5H stickers.
  6. Set Up Final Equation: We are told that in the end, Dalia has 4848 more stickers than Huda. So, 0.8D+0.5H=0.5H+0.2D+480.8D + 0.5H = 0.5H + 0.2D + 48.
  7. Simplify Equation: Simplify the equation: 0.8D+0.5H=0.5H+0.2D+480.8D + 0.5H = 0.5H + 0.2D + 48 becomes 0.6D=480.6D = 48.

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