a woman spends 3 / 5 of her money on food and 1 / 2 of the remainder for a book. if she had 32 left, how much had she at first
To solve this problem, let's break it down step by step.
Step 1: Determine the fraction of money the woman spends on food.
The woman spends 3/5 of her money on food. This means that she spends 3 parts out of a total of 5 parts of her money on food.
Step 2: Calculate the remaining money after spending on food.
Since she spends 3/5 of her money on food, she has 2/5 of her money remaining. This can be calculated by subtracting 3/5 from 1:
1 - 3/5 = 5/5 - 3/5 = 2/5
So, she has 2/5 of her money remaining after spending on food.
Step 3: Determine the fraction of the remaining money spent on a book.
The woman spends 1/2 of the remaining money on a book. Since she had 2/5 of her money remaining after spending on food, she spends 1/2 of 2/5 on a book:
1/2 * 2/5 = 2/10 = 1/5
So, she spends 1/5 of her money on a book.
Step 4: Calculate the remaining money after spending on a book.
Since she spends 1/5 of her money on a book, she has 4/5 of her money remaining. This can be calculated by subtracting 1/5 from 1:
1 - 1/5 = 5/5 - 1/5 = 4/5
So, she has 4/5 of her money remaining after spending on a book.
Step 5: Determine the value of 4/5 of her money.
We are told that she has $32 left, which represents 4/5 of her money. To find out how much 1/5 represents, we divide $32 by 4:
$32 / 4 = $8
Now that we know the value of 1/5, we can find the value of the entire amount by multiplying it by 5:
$8 * 5 = $40
Therefore, the woman had $40 at first.
original amount --- x
spent on food = 3x/5, leaving her with 2x/5
she spends 1/2 of that on a book, leaving her with (1/2)(2x/5) or x/5
so x/5 = 32
x = ....