Daniel and Lauren are painters. Daniel is faster than Lauren
by 2 hours. Occasionally they work with Ian, who paints very
quickly, as fast as Daniel and Lauren together. One day, they
had to paint a room. Lauren and Daniel started to paint first
for 1 hour, then Ian arrived an all 3 of them finished painting
after another 3.5 hours. How long will it take each of them
to paint the room individually?
Make this chart to include all 5 situations mentioned:
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel
Lauren
Ian
D&L
D&L&I
Daniel is faster than Lauren by 2 hours.
Let the time it takes Lauren to paint 1 room be L hours.
Therefore the time it takes Daniel to paint 1 room is L-2.
So we fill in 1 each for the number of rooms each can paint
and L and L-2 for their times
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel 1 L-2
Lauren 1 L
Ian
D&L
D&L&I
We fill in their rates in rooms/hour by dividing rooms by
hours:
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel 1 L-2 1/(L-2)
Lauren 1 L 1/L
Ian
D&L
D&L&I
...Ian, who paints...as fast as Daniel and Lauren together
So his rate is the sum of their rates, so we add their rates
and put that for Ian's rate and 1 for the number of rooms
he could paint at this rate.
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel 1 L-2 1/(L-2)
Lauren 1 L 1/L
Ian 1 1/(L-2)+1/L
D&L
D&L&I
Lauren and Daniel started to paint first for 1 hour,
So we add their rates and put that for the rate for
D&L and also put 1 hour for the time for D&L
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel 1 L-2 1/(L-2)
Lauren 1 L 1/L
Ian 1 1/(L-2)+1/L
D&L 1 1/(L-2)+1/L
D&L&I
Next we get the fraction of a room that they painted
during that 1 hour, by multiplying their rate by
their time.
then Ian arrived an all 3 of them finished painting after another 3.5 hours
So we fill in 3.5 hours for the time for D&L&I, and add the rates for
Ian and D&L. They are the same rates so we just multiply it by 2:
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel 1 L-2 1/(L-2)
Lauren 1 L 1/L
Ian 1 1/(L-2)+1/L
D&L 1/(L-2)+1/L 1 1/(L-2)+1/L
D&L&I 3.5 2(1/(L-2)+1/L)
Then we fill in the fraction of the room they
painted after Ian joined in, by multiplying the
rate on the bottom row by the time:
Number
of rooms Painting
painted Time rate
or fraction painting in
thereof in hours rooms/hr
------------------------------------------------
Daniel 1 L-2 1/(L-2)
Lauren 1 L 1/L
Ian 1 1/(L-2)+1/L
D&L 1/(L-2)+1/L 1 1/(L-2)+1/L
D&L&I 2[3.5(1/(L-2)+1/L] 3.5 2(1/(L-2)+1/L)
Then the equation comes from adding the fractions of
a room painted in the last two lines and setting that
equal to 1 room:
1/(L-2)+1/L + 2[3.5(1/(L-2)+1/L)] = 1
1/(L-2)+1/L + 7(1/(L-2)+1/L) = 1
8(1/(L-2)+1/L) = 1
8/(L-2)+8/L = 1
Multiply through by the LCD of L(L-2)
8L + 8(L-2) = L(L-2)
8L + 8L - 16 = L² - 2L
16L - 16 = L² - 2L
0 = L² - 18L + 16
Solve that by the quadratic formula: Get L = 17.062 hrs.
approximately
(we also get L = 0.938 which is extraneous.)
So it takes Lauren 17.062 hrs to paint 1 room.
Since it takes 2 hours less for Daniel to paint 1 room,
So it takes Daniel 15.062 hrs to paint 1 room.
Finally we must calculate Ian's time to paint 1 room.
Ian's time is 1 room divided by his rate:
1 ÷ [1/(L-2)+1/L] =
1 ÷ [1/(17.063-2)+1/17.063] =
1 ÷ 0.125
8
So it takes Ian 8 hours to paint 1 room.
Edwin
