Let's solve this step by step:
a) To find the fraction of money Judy had left, we need to subtract the fractions she spent from 1.
Fraction spent on shoes = 1/6
Fraction spent on bag = 2/5
Total fraction spent = 1/6 + 2/5
To subtract this total fraction spent from 1, we need to find a common denominator for the fractions. The least common multiple (LCM) of 6 and 5 is 30.
1 can be written as 30/30.
So, the fractions spent can be written as:
Fraction spent on shoes = 5/30
Fraction spent on bag = 12/30
Total fraction spent = 5/30 + 12/30 = 17/30
Fraction of money left = 1 - 17/30
To subtract fractions, we need a common denominator. The denominator is already 30, so we can directly subtract the numerators:
Fraction of money left = (30 - 17)/30 = 13/30
Therefore, Judy has 13/30 of her money left.
b) We are given that Judy had $26 left after spending the money on shoes and a bag.
Let's assume the total money Judy had at first as x dollars.
According to the given information, Judy spent 1/6 of her money on shoes and 2/5 of it on a bag.
So, the money spent on shoes = (1/6) * x
The money spent on the bag = (2/5) * x
The total money Judy spent is the sum of the money spent on shoes and the money spent on the bag:
(1/6) * x + (2/5) * x = $26
To solve this equation, let's find a common denominator:
(5/30) * x + (12/30) * x = $26
(17/30) * x = $26
To isolate x, we need to get rid of the fraction by multiplying both sides of the equation by the reciprocal of (17/30):
x = ($26) * (30/17)
Now, we can calculate the value of x:
x = ($780) / 17
Using a calculator, the value of x ≈ $45.88 (rounded to the nearest cent).
Therefore, Judy had approximately $45.88 at first.