This is the solution to the algebra work word problem # 4 which read: "Joe and Jim paint a house. Joe can paint it alone in 5 days, Jim in 8 days. They start to paint it together but after 2 days Jim stops. How long will Joe finish it alone?"
Joe takes 5 days
Jim 8
to paint a house
So Joe can do 1/5 of his job in one day and Jim can do 1/8th of his job in 1 day. If you just take Joe for example you can see where I'm heading on solving this problem. Joe can do 1/5th of his job in one day. So how long would he finish his job -- well besides knowing the answer already -- you would set it up as x/5 = 1. 1 being when the job is complete not just a "fraction" of his work being done. So the answer is 5 of course 5/5 = 1. You use this exact same method to solve the problem for both Joe and Jim.
**********Correction************ I realized I solved the problem before if they both continued working the job, but this problem states JIM stops working in 2 days. So Joe does x/5 part of his job in 2 days? 2/5 and Jim does x/8 part of his job in 2 days? 2/8, and Joe keeps working. So the equation would be
2/5 + 2/8 + x/5 = 1 multiply by 40 and get
16 + 10 + 8x = 40
26 + 8x = 40
8x = 40 - 26
8x = 14
x = 14/8 or 1 and 3/4ths so it would take 1 and 3/4ths days more for Joe to finish the job alone..
Saturday, November 14, 2009
Work Problem 4 Solution
Wednesday, October 7, 2009
Mixture Problem # 6 Solution
This is the solution to the algebra mixture word problem # 7 which read:
"If a merchant has two types of tea, one worth $2.70 per kilogram and the other worth $3.00 per kilogram, how many kilograms of each type should the merchant use in order to produce 30 kilograms of a blend that is worth $2.95 per kilogram?"
Even though this involves money we treat it like a mixture problem.
You got to unknowns but if you look at it as having x = 1 amount and y = 30 -x then you just have x, and 30 -x so...
Its the x amount * the 270 cent worth tea plus the 30 - x amount * the 300 cent worth tea is going to equal 30 * 295 cent blend
270 * x + 300(30-x) = 30(295)
270x + 9000 - 300x = 8850
-30x = -150
30x = 150
x = 150/30 = 5
so 5 kilograms of the 270 cent or $2.70 tea and 30 -5 or 25 kilograms of the 300 cent or $3.00 tea would need to be mixed together to make 30 kilograms of the $2.95 cent tea.
check 270 * 5 + 300 * 25 = 30*295
1350 + 7500 = 8850
8850 = 8850
Number Problem 7 Solution
This is the solution to the algebra number word problem 7 which asked "If the Numerator and Denominator of a certain fraction are both increased by 3, the resulting fraction equals 2/3. If the Numerator and Denominator are both decreased by 2, the resulting fraction equals 1/2. Determine the fraction."
Ok so you have 2 unknowns --> x the numerator and y the denominator. You can solve 2 unknowns with 2 equations. They give you 2 equations.
(x+3)/(y+3) = 2/3 and
(x-2)/(y-2) = 1/2
take the (x+3)/(y+3) = 2/3 and cross multiply and get:
2(y+3) = 3(x+3) which is
2y + 6 = 3x + 9
Now take the second equation: (x-2)/(y-2) = 1/2 and cross multiply
2(x-2) = 1(y-2) which makes
2x - 4 = y - 2 now solve for y
y = 2x - 2
ok now plug this back into the 1st equation for y which we broke down to :
2y + 6 = 3x + 9 so you get 2(2x-2) + 6 = 3x + 9 solve for x
4x - 4 + 6 = 3x+ 9
x= 7 ok now plug 7 into y = 2x - 2 and get y = 2(7) -2 = 12
so the numerator is 7 and the denominator is 12 = 7/12
to check add 3 to the top and bottom and get 10/15 = 2/3
and subtract 2 from the top and bottom and get 5/10 = 1/2
Saturday, September 12, 2009
Time and Distance Problem 10 Solution
This is the solution to the algebra time and distance word problem # 10 which was submitted under the ask a question post anonymously which read..
"Junior's boat will go 15 miles per hour in still water. If he can go 12 miles downstream in the same amount of time as it takes to go 9 miles upstream then what is the speed of the current?"
distance = rate X time ... d = r*t we know the speed of the boat in still water 15 mph
we know he can go 12 miles downstream in the same time he can go 9 miles upstream-- so times are equal --- so since time is equal we can set the distances and rates equal to one another...
d1/r1 = d2/r2
The rate downstream will be 15 + x and upstream will be 15 - x so we have:
12/(15+x) = 9 /(15-x)
cross multiply and you get 12(15-x) = 9 (15+x)
180 - 12x = 135 + 9x
45 = 21x
x = 45/21 = 15/7 This is the answer the current is 15/7 or 2 and 1/7ths mph..
Tuesday, August 25, 2009
Time and Distance Problem 9 Solution
This is the solution for the time and distance algebra word problem #9 that read:
"if I walked 9 laps around a track. One lap is 1312 ft. I walked at a rate of 4 miles an hour. how many minutes did it take me to walk 9 laps?"
The easiest way to think about this problem is to think about an easier problem and then copy the steps. What if the problem read:
if I walked 9 laps around a track -- 1 lap is 1 mile and i walked at a rate of 4.5 miles and hour. how many minutes did take me to walk 9 laps?
well the total distance would be the length of 1 lap: 1 mile * the # of laps which is 9 --> 9 * 1 = 9 miles
the rate was 4.5 miles an hour -- distance = rate * time
so 9 miles = 4.5miles/hour* time
9 miles/4.5 miles/hour = time or 2 hours would be how long it would take--->
So this is not a hard math problem the arithmetic is what's hard in this problem, but if you just follow the same steps you will get the answer easily enough. So, again we have:
"if I walked 9 laps around a track. One lap is 1312 ft. I walked at a rate of 4 miles an hour. how many minutes did it take me to walk 9 laps"
ok so take 9 * 1312 1st to get the total distance in feet which is : 11808
4 miles an hour would be (5280 ft in a mile) 5280 * 4 = 21120 ft/hour
But we dont want hours we want minutes-- there are 60 minutes in an hour
so (21120 ft*1hour)/(hour*60minutes) --> you're multiplying by 1hour/60 minutes. so divide by 60..
362ft/minute --- the hours(1 hour = hour) cancel out..
*** Correction -- user Donaldson pointed out I made a division error -- I said 362 feet should be 352 feet/min...*** Thanks!
ok so now we have a total distance in feet of 11808 feet -- divided by the rate of 352 ft/minute
11808 ft/(352 ft/minute) = 11808 ft *minute/ 352 ft
the ft cancel out you divide 11808 by 352 and get 33.54545 = 33.55 minutes --> which is a good enough answer but if you want it more exact:
.545454 of 60 seconds would be 60 *.5454 = 32.72 seconds == approx 33 seconds --- so 32 minutes 33 seconds would be a more recognizable (as far as common usage of time) answer.
You can kind of double check the answer by making sure its reasonable when you see that the total distance of 11808 feet that you walked is roughly 2 miles and your dividing by 4 miles an hour=== 1/2 an hour or 30 minutes approximately
Sunday, August 2, 2009
Plane Geometric Figure Problem 1 Solution
This is the solution to the Plane Geometric Figure Problem 1 which asked you to "find the dimension of a rectangle whose length is 9 less than twice its width if the perimeter is 120 cm."
In a rectangle 2 times the width plus to times the length equals the total perimeter.
Let the width equal x
Then the length is equal to 2x - 9 which means "9 less than twice its width"
Remember 2 times the width plus to times the length equals the perimeter so..
2(x) + 2(2x - 9) = 120
which is..
2x + 4x - 18 = 120
6x - 18 = 120
6x = 120 + 18
6x = 138
x = 138/6
x = 23
Since we let x stand for the width then the width of the rectangle is 23 cm and the length is 2(23) - 9 = 46 - 9 = 37
So the width is 23 cm and the length is 37 cm
Check 2(23) + 2(37) = 120
46 + 74 = 120
120 = 120
Age Problem 4 Solution
This is the solution to the algebra age problem # 4... "A man is 27 years older than his son and 10 years from now, he will be twice as old as his son. how old is each now?"
Let x = the son and x + 27 = the man(father)
in 10 years the son will be x + 10 and the father x + 27 + 10 or x + 37 years old. So if he will be twice as old as his son then "twice" the sons age then will "equal" each other.
So, 2(x+10) = x + 37
multiply by 2 on the left side
2x + 20 = x + 37
x = 17 This makes sense since 10 + 17 = 27 and 17 + 37 = 54 which is twice the sons age 10 years from now. So the son is 17 and the father is 17 + 27 or 44 years old.
