Judy spent 1/6 of her money on a pair of shoes and 2/5 of it on a bag. She then had $26 left.
a) What Fraction of her money did she have left?
b) How much money did she have at first?
Taking x = the initial amount of money
=> x - (1/6)x - (2/5)x = 26
=> x - (5/30)x - (12/30) = 26
=> x - (17/30)x = 26
a) She spent a total of (17/30) of her money, so she had (13/30) of it left.
b)
=> x - (17/30)x = 26
=> x(1 - 17/30) = 26
=> (13/30)x = 26
=> x = 26*(30/13)
= 2*30
= $60
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.
To find the answer to these questions, we can use a step-by-step approach.
a) What fraction of her money did she have left?
To find the fraction of her money that Judy had left, we need to subtract the fractions representing the amount she spent on the shoes and bag from 1 (which represents the whole amount of money she had initially).
Step 1: Determine the fraction spent on the shoes:
Judy spent 1/6 of her money on a pair of shoes.
Step 2: Determine the fraction spent on the bag:
Judy spent 2/5 of her money on a bag.
Step 3: Add the fractions together:
1/6 + 2/5 = (5/30) + (12/30) = 17/30.
Step 4: Subtract the sum of the spent fractions from 1:
1 - 17/30 = 13/30.
Therefore, Judy had 13/30 of her money left.
b) How much money did she have at first?
To find out how much money Judy had initially, we can work backwards using the fraction of money she had left.
Step 1: Write the fraction as an equation:
13/30 = $26 (the amount left).
Step 2: Solve for the whole amount of money:
Let's assume the initial amount of money Judy had is x. To solve the equation, we can set up a proportion: 13/30 = $26/x.
Cross-multiplying, we get: x = (30 * $26) / 13.
Step 3: Calculate the value of x:
x = $780 / 13 = $60.
Therefore, Judy initially had $60.