Beverly sells mangoes. She sold half her mangoes plus half a mango last Monday. Then she sold half her remaining mangoes plus a mango last tuesday. Again she sold half of her remaining mangoes plus half a mango on wednesday. She then sold the remaining mangoes on thursday. How many mangoes did beverly have at the beginning?
To find the number of mangoes Beverly had at the beginning, we need to work backward from Thursday to Monday. Let's break down the problem step by step:
1. Thursday: Beverly sold the remaining mangoes. Since no mangoes were left after Thursday, we can determine the number of mangoes she sold on Thursday by working backward.
2. Wednesday: Beverly sold half of her remaining mangoes plus half a mango. This means half of the mangoes she had on Wednesday plus half a mango equals the number of mangoes she sold on Thursday.
3. Tuesday: Beverly sold half of her remaining mangoes plus a mango. This means half of the mangoes she had on Tuesday plus one mango equals the number of mangoes she had left on Wednesday.
4. Monday: Beverly sold half her mangoes plus half a mango. This means half of the mangoes she had on Monday plus half a mango equals the number of mangoes she had left on Tuesday.
Now, let's solve the problem step by step:
Step 4: Monday
- Let's assume the number of mangoes Beverly had on Monday is x.
- She sold half of x, which is x/2.
- She also sold half a mango.
- Therefore, she had x/2 - 1/2 remaining mangoes on Tuesday.
Step 3: Tuesday
- Beverly had x/2 - 1/2 remaining mangoes on Tuesday.
- She sold half of the remaining mangoes, which is (x/2 - 1/2) / 2 = (x/4 - 1/4).
- She also sold one mango.
- Therefore, she had (x/4 - 1/4) - 1 remaining mangoes on Wednesday.
Step 2: Wednesday
- Beverly had (x/4 - 1/4) - 1 remaining mangoes on Wednesday.
- She sold half of the remaining mangoes, which is [(x/4 - 1/4) - 1] / 2 = (x/8 - 1/8 - 1/2).
- She also sold half a mango.
- Therefore, she had (x/8 - 1/8 - 1/2) - 1/2 remaining mangoes on Thursday.
Step 1: Thursday
- Beverly had (x/8 - 1/8 - 1/2) - 1/2 remaining mangoes on Thursday.
- She sold the remaining mangoes, which is [(x/8 - 1/8 - 1/2) - 1/2] = 0.
- This means that (x/8 - 1/8 - 1/2) - 1/2 = 0.
Now, let's solve for x:
(x/8 - 1/8 - 1/2) - 1/2 = 0
(x/8 - 1/8 - 1/2) = 1/2
(x/8 - 1/8) = 1/2 + 1/2
(x/8 - 1/8) = 1
Combining like terms:
(x/8 - 1/8) = 1
(x - 1)/8 = 1
Multiply both sides by 8:
(x - 1) = 8
Add 1 to both sides:
x = 9
Therefore, Beverly had 9 mangoes at the beginning.