+0

# fractions

+2
155
2
+318

Bart had 57 less madeleines than Jeff. Bart saved 3/5 of his madeleines while Jeff spent 2/3 of the madeleines. Given that the amount Bart spent was 1/4 of Jeff’s savings, how much did Jeff spend?

Jul 7, 2021

#1
+318
+5

Let the number of madeleines with Jeff be 'j' and the number of madeleines with Bart be 'b'.

Then, if Bart had 57 fewer madeleines than Jeff: $$b = j - 57 …1$$

Given, Bart saved 3/5 of his madeleines.

He must have spent the rest:
$$1$$$$-$$$${3 \over 5}\frac{}{}$$$$= {2\over 5}\frac{}{}$$

Bart's spending =$${2 \over 5}b\frac{}{}$$

If Jeff spent 2/3 of his madeleines.

He must have saved the rest:

$$1 -$$$${2 \over 3}={}{}$$$${1 \over 3}\frac{}{}$$

Jeff savings=$${1 \over 3}\frac{}{}$$

Given: the amount Bart spent was 1/4 of Jeff’s savings.

Thus,

$${2 \over 5}{}{}$$$$b$$$$=$$$${1 \over 4}{}{}$$$$($$$${1 \over 3}j{}{}$$$$)$$

Cross multiply:

2 * 4 * 3 * b = 1 * 5 * j

24b = 5j ...2

Solve using equation 1 and equation 2:

$$b = j - 57 …1$$
$$24b=5j …2$$

$$24(j-57) = 5j$$

$$24j-1368=5j$$

$$19j=1368$$

$$j = {1368 \over 19}{}^{}=72$$

The question is how much did Jeff spend:

Given, Jeff spent 2/3 of the original value: $${2 \over 3}{}{} * 72 = 48$$

Jul 7, 2021

#1
+318
+5

Let the number of madeleines with Jeff be 'j' and the number of madeleines with Bart be 'b'.

Then, if Bart had 57 fewer madeleines than Jeff: $$b = j - 57 …1$$

Given, Bart saved 3/5 of his madeleines.

He must have spent the rest:
$$1$$$$-$$$${3 \over 5}\frac{}{}$$$$= {2\over 5}\frac{}{}$$

Bart's spending =$${2 \over 5}b\frac{}{}$$

If Jeff spent 2/3 of his madeleines.

He must have saved the rest:

$$1 -$$$${2 \over 3}={}{}$$$${1 \over 3}\frac{}{}$$

Jeff savings=$${1 \over 3}\frac{}{}$$

Given: the amount Bart spent was 1/4 of Jeff’s savings.

Thus,

$${2 \over 5}{}{}$$$$b$$$$=$$$${1 \over 4}{}{}$$$$($$$${1 \over 3}j{}{}$$$$)$$

Cross multiply:

2 * 4 * 3 * b = 1 * 5 * j

24b = 5j ...2

Solve using equation 1 and equation 2:

$$b = j - 57 …1$$
$$24b=5j …2$$

$$24(j-57) = 5j$$

$$24j-1368=5j$$

$$19j=1368$$

$$j = {1368 \over 19}{}^{}=72$$

The question is how much did Jeff spend:

Given, Jeff spent 2/3 of the original value: $${2 \over 3}{}{} * 72 = 48$$

apsiganocj Jul 7, 2021
#2
+857
+2

Let $a_1$ equal Bart's madeleines, and let $a_2$ equal Jeff's madeleines.

$a_1 + 57 = a_2$

$\left(1-\frac{3}{5}\right) a_1 = \frac{1}{4} \cdot \left(1-\frac{2}{3}\right) a_2$

$\frac{2}{5} a_1 = \frac{1}{12} a_2$

$24 a_1 = 5 a_2$

$24 a_1 = 5(a_1 + 57)$

$24 a_1 = 5 a_1 + 5 \cdot 57$

$19 a_1 = 5 \cdot 57$

$a_1 = \frac{5 \cdot 19 \cdot 3}{19}$

$a_1 = 15$

$a_2 = 15 + 57 = 72$

$\frac{2}{3} a_2 = \frac{2}{3} \cdot 72 = 24 \cdot 2 = \boxed{48}$