Emma started with a certain amount of money. She spent 2/5 of her money. She earned €5 the next day. Later, she lost 3/7 of her money, but she still had €20 left. How much did she start with?
had (3x/5 + 5)
(4/7)(3x/5 + 5) = 20
The answer at the back of the book is €50...
(4/7)(3x/5 + 5) = 20
4(3x +25) = 20 * 7 * 5
3x + 25 = 5 * 7 * 5 = 175
3 x = 150
x = 50
I guess the book is correct.
To find out how much money Emma started with, let's go step by step.
We know that Emma started with a certain amount of money.
Let's represent the amount she started with as "x" euros.
Emma spent 2/5 of her money, which means she had 3/5 of her money remaining.
After spending, Emma had (3/5)x euros left.
The next day, Emma earned 5 euros. Adding this to what she had before, Emma had (3/5)x + 5 euros.
Afterward, Emma lost 3/7 of her money, which means she had 4/7 of her money remaining.
After losing, Emma had (4/7)((3/5)x + 5) euros left.
We're told that she still had 20 euros left, so we can set up an equation:
(4/7)((3/5)x + 5) = 20
To simplify, multiply 4/7 by ((3/5)x + 5):
(4/7) * ((3/5)x + 5) = 20
Multiply both sides by 7:
4 * ((3/5)x + 5) = 140
Distribute the 4:
(12/5)x + 20 = 140
Subtract 20 from both sides:
(12/5)x = 120
To isolate x, divide both sides by (12/5), which is the same as multiplying by the reciprocal:
x = (120)/(12/5)
To divide by a fraction, invert the fraction and multiply:
x = (120)(5/12)
Simplify:
x = 50
Therefore, Emma started with €50.