a) A woman is twice as old as her daughter. Five years ago the sum of their age was 62. What are their present ages.
b) A bottle is half full. After adding 275 mL to the bottle it is three quarters full. How much does the bottle hold when full?
A woman is twice as old as her daughter mean:
W = 2 D
Five years ago wooman was W - 5 yrs , daughter was D - 5 yrs
Five years ago the sum of their age was 62 mean:
W - 5 + D - 5 = 62
W - 10 + D = 62 Add 10 to both sides
W - 10 + D + 10 = 62 + 10
W + D = 72
Replace :
W = 2 D in this equation
2 D + D = 72
3 D = 72 Divide both sides by 3
D = 72 / 3 = 24 yrs
W = 2 D = 2 * 24 = 48 yrs
b)
V = Volume of a bottle
V / 2 + 275 = ( 3 / 4 ) V Subtract V / 2 to both sides
V / 2 + 275 - V / 2 = ( 3 / 4 ) V - V / 2
275 = ( 3 / 4 ) V - V / 2
275 = ( 3 / 4 ) V - ( 2 / 4 ) V
275 = ( 1 / 4 ) V Multiply both sides by 4
275 * 4 = ( 1 / 4 ) V * 4
1100 = V
V = 1100 mL
a) To solve this problem, let's assign variables to represent the ages of the woman and her daughter. Let's say the daughter's age is x. Since the woman is twice as old as her daughter, the woman's age would be 2x.
Five years ago, the daughter's age would have been x - 5, and the woman's age would have been 2x - 5. Given that the sum of their ages five years ago was 62, we can write the equation:
(x - 5) + (2x - 5) = 62
3x - 10 = 62
3x = 62 + 10
3x = 72
x = 72 / 3
x = 24
So, the daughter's current age is 24, and the woman's current age is twice that, which is 2 * 24 = 48.
Therefore, the woman is 48 years old, and her daughter is 24 years old.
b) Let's solve this problem by setting up an equation. We know that the bottle is initially half full, so let's represent the full capacity of the bottle as x mL.
When 275 mL is added, the bottle becomes three quarters full, which means it is now 3/4 * x mL.
We can set up the equation:
(1/2) * x + 275 = (3/4) * x
To solve for x, we can first multiply both sides of the equation by 4 to eliminate the fraction:
4 * (1/2 * x + 275) = 4 * (3/4 * x)
2x + 1100 = 3x
Next, we can subtract 2x from both sides to isolate the variable:
2x - 2x + 1100 = 3x - 2x
1100 = x
Therefore, the bottle holds 1100 mL when full.