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) $.

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)