Alice had twice as much money as Peter. After Alice spent 1/6 of her money
and Peter spent
2/3 of his money, they had a total of $720 left.
(a) How much money did Alice and Peter have altogether at first?
(b) Peter spent 3/4 of his remaining money on a bag. What fraction of his
original amount of money did he spend on the bag?
(Give your answer in its simplest form.)
I will solve part (a) only.
Let x be the amount Peter had at first, in dollars.
Then Alice had 2x dollars.
Write equation for the total remaining money after spending, as you read the problem
(5/6)(2x) + (1/3)x = 720 dollars.
To solve, multiply both sides of the equation by 6 and simplify
10x + 2x = 6*720
12x = 6*720
x = (6*720)/12 = 6*60 = 360.
Peter had $360 at first; Alice had twice of it, i.e. $720 dollars.
Altogether, they had 360 + 720 = 1080 dollars, at first.
(a)
Let the money Peter had be y, therefore, Alice had 2y.
Total money= (y+2y)=3y
Alice spent (1/6 of 2y)=1/3y
Peter spent (2/3 of y)=2/3y
Total money spent= (1/3y+2/3y)=y
Total money left= (3y-y)=2y
2y=$720 (divide both sides by 2)
y=$360
Peter had $360
Alice had (2×360)=$ 720
Hence, total money is (360+720) =$1080
(b)
After Peter spent 2/3, the remaining fraction of money is (1-2/3)=1/3
(1/3 of $360) =$120
Money spent on the bag =(3/4 of $120)=$90
Therefore, fraction of the amount spent on bag over the original amount
=90/360
=1/4
To solve this problem, we can use algebraic equations. Let's assign variables to the unknowns. Let's say x represents the amount of money Peter had initially, and y represents the amount of money Alice had initially.
(a) We are given that Alice had twice as much money as Peter. So we can write the equation y = 2x.
After Alice spent 1/6 of her money, she has 5/6 of her original amount left. And after Peter spent 2/3 of his money, he has 1/3 of his original amount left.
The sum of the remaining money is $720. So we can write the equation (5/6)y + (1/3)x = 720.
To solve these two equations simultaneously, we can substitute the value of y from the first equation into the second equation:
(5/6)(2x) + (1/3)x = 720
(10/6)x + (1/3)x = 720
(5/3)x + (1/3)x = 720
(6/3)x = 720
2x = 720
x = 360
Substitute the value of x back into the first equation to find y:
y = 2(360)
y = 720
Therefore, Peter initially had $360, and Alice initially had $720.
(b) Peter spent 3/4 of his remaining money on a bag. After spending that amount, he has 1/4 of his original amount left. Since we know that he initially had $360, we can find out how much he spent on the bag:
Amount spent on the bag = (3/4) * $360 = $270.
To find the fraction of his original amount of money spent on the bag, we divide the amount spent by his initial amount:
Fraction spent = $270 / $360
Simplifying the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 90 in this case:
Fraction spent = (270/90) / (360/90) = 3/4
Therefore, Peter spent 3/4 of his original amount of money on the bag.