How much more money?
The ratio of Beatriz’s savings to Hannah’s savings is 4 : 7. The ratio of Hannah’s savings to Julia’s savings is 3 : 5. The three girls have a total savings of $136. How much more money has Julia saved than Beatriz? Explain how you got your answer.
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To find out how much more money Julia has saved than Beatriz, we need to determine the amount of money each girl has saved individually.
Let's assign variables to represent the amounts each girl has saved. Let B represent Beatriz's savings, H represent Hannah's savings, and J represent Julia's savings.
According to the given information, the ratio of Beatriz's savings to Hannah's savings is 4:7, which means for every $4 Beatriz saves, Hannah saves $7. Similarly, the ratio of Hannah's savings to Julia's savings is 3:5, meaning for every $3 Hannah saves, Julia saves $5.
Now, we can set up a system of equations to represent the given information:
Equation 1: B/H = 4/7
Equation 2: H/J = 3/5
Equation 3: B + H + J = 136
To make it easier, let's get rid of the fractions by multiplying both sides of Equation 1 by 7 and both sides of Equation 2 by 5:
Equation 1 (modified): 5B = 28H
Equation 2 (modified): 3H = 5J
Now we can use these modified equations to eliminate variables and solve for their values.
1) Multiply Equation 1 (modified) by 3 and Equation 2 (modified) by 28 to make the coefficients of B and H equal:
15B = 84H
84H = 140J
2) Since both equations have 84H, we can set them equal to each other:
15B = 140J
3) Rearrange the equation to isolate B:
B = (140J) / 15
4) Substitute this expression for B in Equation 3:
(140J) / 15 + H + J = 136
5) Simplify the equation:
9J + 15H + 5J = 136
6) Combine like terms:
14J + 15H = 136
Now we have an equation with only two variables, J and H. We need another equation to solve for the individual amounts saved by each girl.
7) We can use Equation 2 to eliminate H:
3H = 5J
8) Rearrange the equation to isolate H:
H = (5J) / 3
9) Substitute this expression for H in Equation 7:
14J + 15((5J) / 3) = 136
10) Simplify and solve for J:
14J + (75J) / 3 = 136
Multiply through by 3 to eliminate fractions:
42J + 75J = 408
117J = 408
J ≈ 3.49
Now we know Julia has saved approximately $3.49.
11) Substituting J back into Equation 2:
H = (5 * 3.49) / 3 ≈ 5.15
Hannah has saved approximately $5.15.
12) Substituting J and H back into Equation 3:
B + 5.15 + 3.49 = 136
B ≈ 127.36
Beatriz has saved approximately $127.36.
Finally, to find out how much more money Julia has saved than Beatriz, subtract Beatriz's savings from Julia's savings:
Julia's savings - Beatriz's savings ≈ $3.49 - $127.36 = -$123.87
Therefore, Julia has saved approximately $123.87 less than Beatriz.