Question
Rowena can paint a room in 14 hours, while Ruby can paint it in 6 hours. If Rowena paints for x hours and Ruby paints for y hours, they will finish half of the painting, while if Rowena paints for y hours and Ruby paints for x hours they will paint the whole room. Find the ordered pair (x,y)
Answer:
(x, y) = (231/40, 21/40)
where x = 231/40
y = 21/40
Step-by-step explanation:
From the question, we are told:
Rowena can paint a room in = 14 hours
Ruby can paint it in = 6 hours.
This means
In one hour.
Rowena can paint = 1/14 of the room
Ruby can paint =1/6 hour of the room
From the question, we are told that :
If Rowena paints for x hours and Ruby paints for y hours, they will finish half of the painting
This is represented mathematically as:
(1/14)(x) + ( 1/6)(y) = 1/2...... Equation 1
Also we were told that:
if Rowena paints for y hours and Ruby paints for x hours they will paint the whole room
(1/14)(y) + (1/6)(x) = 1 .......... Equation 2
Bringing the two equations together, we have:
(1/14)(x) + ( 1/6)(y) = 1/2 ....... Equation 1
(1/14)(y) + (1/6)(x) = 1 ............ Equation 2
We find the Lowest common multiple of the numerator 14 and 6 = 42. Hence, we multiply both equations through by 42
3x + 7y = 21 ....... Equation 3
7x + 3y = 42 ......... Equation 4
To solve for x and y in Equation 3 and 4 we would be using the Elimination method
21x + 49y = 147 .......... Equation 5
21x +9y = 126 ........... Equation 6
21x + 49y = 147 .......... Equation 5
-(21x +9y = 126) ........... Equation 6
40y = 21
y = 21/40
To get the value of x , we would substitute 21/40 for y in Equation 3
3x + 7y = 21 ....... Equation 3
3x + 7(21/40) = 21
3x + 147/40 = 21
3x = 21 - 147/40
3x = 693/40
x = (693/40) ÷ 3
x = 693/40 × 1/3
x = 693/120
x = 231/40
Therefore, (x, y) = (231/40, 21/40)
