Final answer:
Hazel originally had 229 cupcakes, which is calculated using the information from the two given scenarios involving ratios and equality after certain numbers of cupcakes are given from Hazel to Kate.
Step-by-step explanation:
To solve for the number of cupcakes Hazel has at first, we need to use the information given about the ratios after certain transactions and the point at which they have equal cupcakes. We are given two scenarios:
- When Hazel gives Kate 49 cupcakes, the ratio of Hazel's to Kate's cupcakes is 3:4.
- When Hazel gives Kate 19 cupcakes, they will have an equal number of cupcakes.
Let's denote the number of cupcakes Hazel has initially as H and the number of cupcakes Kate has initially as K.
From the second scenario, H - 19 = K + 19, implying H - K = 19 + 19, or H - K = 38 (Equation 1).
From the first scenario, we can set up the ratio (H - 49) / (K + 49) = 3/4. Cross-multiplying gives 4(H - 49) = 3(K + 49), which simplifies to 4H - 3K = 343 (Equation 2).
Using simultaneous equations, we can solve for H and K. Multiply Equation 1 by 3, you get 3H - 3K = 114 (Equation 3). Now subtract Equation 3 from Equation 2: 4H - 3H = 343 - 114, thus H = 229.
The answer is that Hazel originally had 229 cupcakes, which is not listed in the options provided, indicating a possible typo in the multiple-choice answers.