Number Problem 15 Solution

A number added to five times its reciprocal is -6. Find the number.



Solution to the problem:




A number = x

Reciprocal of that number would be 1/x

5 times the reciprocal would be  5 *  1/x   =  5/x

A number added to five times its reciprocal would then be x + 5/x

This then is equal to -6.

x + 5/x = = -6

Multiply by x...

x^2 + 5 = -6x

x^2 + 6x + 5 = 0

(x + 1 )(x + 5 )

The answer are -1 and -5.


Check

-1 + 5/-1  = -1 -5 = -6  Correct and

-5 + 5/-5 = -5 -1 = -6  Correct

Plane Geometric Figure Problem 3 Solution

This is the solution for the Algebra plane geometric figure problem as asked by an anonymous visitor which read, "catrina and tom want to buy a rug for a room that s 14 by 15 feet. They want to leave an even strip of flooring uncovered around the edges of the room. How wide a strip if they buy a rug with an area of 110 square feet?"

It's easier to picture this.  Draw a rectangle that's almost a square 14 x 15 feet.  Within in draw another such figure but smaller.   The space between that figure and the 14 x 15 figure is x.  Now the smaller square has an area of 110 square feet.  We now have an equation :


(15 -x ) (14 -x ) = 110

210 + x^2 -14x - 15x = 110

x^2 - 29x + 210 = 110

x^2 -29x + 100 = 0

(x - 25)(x - 4) = 0

x = 4 or 25 however using 25 would result in negative dimensions which does not make sense.  So 4 is the answer

Check:  (15 - 4) (14 - 4) = 110  = 11*10 = 110 

Number Problem 14 Solution

This is the solution for the Algebra number word problem which was asked by an anonymous visitor that read, "55 bowls of the same size capacity of food.
1. everyone gets their own bowl of soup
2. every two gets one bowl of spaghetti to share
3. every three will get one bowl of salad
4. all r required to have their own helping of salad, spaghetti, and soup."

Ok you know the total number of people is x.  Since '" every two gets one bowl of spaghetti to share" then x/2 of the people will have those bowls and since "every three will get one bowl of salad" then x/3 of the people will have those bowls in which they will total 55 bowls. 
So we have:

x + x/2 + x/3 = 55  ** multiply by 6

6x + 3x + 2x = 330

11x = 330

x = 30

So there are 30 bowls for soup, 30/2 = 15 bowls for spaghetti , and 30/3 = 10 bowls for salad. 

Check 30 + 15 + 10 = 55

Finance Problem 4 Solution

This is the solution to the Algebra finance word problem a anonymous user asked, "An administrative assistant orders cellular phones for people in her department. The brand A phones cost $89.95 and the brand b phones $34.95.If she ordered 3 times as many brand b phones as brand a phones at a total cost of $584.40, how many of each did she order?"

Ok to make things easier lets make it cents.  Just remember to divide by 100 when done to get the answer in dollars and cents...

So brand A is 8995 and b  is 3495  which had 3 times as many sold as a and all the combinations of phones totalled  58440.

So 8995x + 3495(3x) = 58440

8995x + 10485x = 58440

19480x = 58440

x = 3   So there were 3 brand A's and 3(3) = 9 brand B's sold.

Finance Problem 3 Solution

This is the solution to the Algebra finance word problem asked by an anonymous user which read, "Alex has made 42 of the 48 payments he owes on his car but is having trouble continuing to make the payment in full. He has made an arrangement with the bank to pay 2/3rds of his monthly payment, rather than the entire payment, every month. How many months will it take Alex to pay off his car loan with this new payment arrangement?"

Ok, 1st find the remaining months :  48 - 42 = 6  So there are 6 months at full payments.  But, you are no longer doing full payments you are doing 2/3s of a payment.  So  2/3 of what would equal out 6 ?  Or

2x/3 = 6

2x = 18

x = 18/2 = 9 .. So it will take Alex 9 months to pay off his car loan with the new payment arrangement..

Number Problem 13 Solution

This is the solution to algebra number problem 13 as asked by an anonymous user: "A maths test contains 10 questions. Ten points are given for each correct answer and three points deducted for an incorrect answer. If Ralph scored 61, how many did he get correct?"

Ok on this problem you can come up with a solution faster by just quick trial an error.  You know that if it only counts 3 points off for each wrong answer and they got a 61 then they had to have at least a 70 before the points were taken off.  If they got a 70 that means they missed 3 problems (3 * 3 = 9)   70 - 9 = 61   .  So the answer is they got 7 problems right and 3 wrong.

Now to set this up using algebra...

10x - 3(10-x) = 61  You are subtracting the wrong answers worth 3  points a piece from the right ones worth 10 a piece.  You don't know how many of each so you are saying there are x amount of 10 valued answers and  (10 - x) number of 3 valued wrong answers. 

10x - (30 - 3x) = 61

10x -30 + 3x = 61

13x = 91  -->   x = 91/13 = 7    --> so there are 7 right 10 valued problems and (10-7) or 3 wrong 3 point negative value ones.....

You could have also set it up like this 10(10 - x) - 3x = 61 and got the same answer....

Age Problem 8 Solution

This is the solution to the algebra  age problem asked by Joshua that read, "A man is 4 years older than his wife and 5 times as old as his son. When the son was born, the age of the wife was six-sevenths that of her husband's age. Find the age of each."
Ok so here we go...

"A man is 4 years older than his wife"
m = w + 4
Now we will subtract 4 from both sides to use this conveniently later ( you will see)
w = (m-4)

"and 5 times as old as his son."
m = 5s

"When the son was born, the age of the wife was six-sevenths that of her husband's age."
w - s = 6/7(m-s)

7(w-s) = 6(m-s) -->
7w - 7s = 6m - 6s --->
7w = 6m - 6s + 7s --->
7w = 6m + s
replace w with (m-4)
7(m-4) = 6m + s
7m - 28 = 6m + s
7m - 6m = s + 28
m = s + 28
replace m with 5s
5s = s + 28
5s - s = 28
4s = 28
s = 7 yrs is the son's age
then
5(7) = 35 yrs is Dad's age
and
35-4 = 31 yrs is Mom's age

Now check work...
"When the son was born, the age of the wife was six-sevenths that of her husband's age."
31 - 7 = 6/7(35 - 7)
24 = 6/7*28
24 = 24;

Number Problem 12 Solution

This is the solution to a number algebra word problem that was asked by Zakeus who said , "The numerator and denominator of a fraction are together equal to 100. Increase the numerator by 18 and decrease the denominator by 16 and the fraction is doubled. What is the fraction?"


Ok you want to set up 2 equations the 1st one is easy.   x + y = 100 so y = 100 - x

The other would be  (x + 18)/(y - 16) = (x/y)*2

(x + 18) y = 2x (y - 16)  Now substitute 100-x for y....

(x + 18)(100 - x) = 2x(100 - x - 16)

100 x + 1800 - x^2 -18x = 2x(84-x)

82x + 1800 -x^2 = 168x - 2x^2  Now combine like terms...

x^2 - 86x + 1800 = 0    

(x - 50)(x - 36) = 0   Now 50/50 is 1 and that's not the fraction we are looking for  ( y = 100 - 50 = 50)

So substituting 36 in for x we have y = 100 - 36 = 64     So the fraction is 36/64  Now to verify this is correct add 18 to the numerator and subtract 16 from the denominator and we get ..

54/48  ...       Ok now to verify if this is double the other equation lets reduce the equations.     36/64  4 is the gcd so we get 9/16     and for 54/48 , 3 is the gcd and we get 18/16   and   9/16 * 2 does equal 18/16, so we have the correct answer....

Time and Distance Problem 13 Solution

This is the solution for a algebra time and distance word problem asked by a anonymous user: "Two trains left a station at the same time. One traveled north at a certain speed and the other traveled south at twice that speed. After 3 hours , the trains were 400 miles apart. How fast was each train traveling?"


The speed of the 1st train is x and the 2nd train is twice that or 2x.  Ok the trick here is lets say someone is going 50 mph north and the other person is going twice that 100 the opposite direction.   This distance separating them would be the same thing as if you added their speeds and it was going one direction -- in other words if someone was just going 150mph they would cover the same ground.    Also rate multiplied by time equals distance.  We know the time is 3 hours and the distance is 400.  So

(x + 2x) (3) = 400

3x(3) = 400

9x = 400

x = 400/9  = 44 and 4/9ths  mph is the 1st train and 88 and 8/9ths (800/9) mph is the 2nd train.

Age Problem 7 Solution

This is the solution to a algebra age word problem as asked by a anonymous user: "Nona is one-third as old as her mother. Five years ago, she was only one-fifth of the age of her mother. How old is Nona now? "


It is easier to set this up with two unknowns say x and y and have 2 equations.

The 1st equation would be:  x = 1/3(y)  or x = y/3   since Nona is currently 1/3 as old as her mother.  Now 5 years ago she was only 1/5th the mothers age so the other equation would be..

x -5 = 1/5(y - 5)   multiply by 5..

5x - 25 = y - 5 so

y = 5x - 20  .. plug this back into the 1st equation and you get..

x = (5x - 20)/3  - -multiply by 3

3x = 5x - 20  so 2x = 20  and x = 10

So Nona is 10 years old currently and the mom is 30.   To check this go 5 years back... Nona would be 5 and the mother would be 25   ------   5/25 = 1/5ths of her mothers age...

Mixture Problem 7 Solution

This is the solution to a new algebra mixture problem as asked by Califax who asked, "A 50-lb solution of acid and water is 20% (by weight). How much pure acid must be added to this solution to make it 30% acid?"

1/5 = 20 %  .....  3/10 = 30 %       x = the unknown amount in lbs we want to add of  100% or 1/1 or 1 or just x of pure acid.

50(1/5) + x = 3/10 (50 + x)  multiply by 10...

100 + 10x = 3(50 + x)

100 + 10x = 150 + 3x

7x = 50  =   50/7  = 7 and 1/7th lbs of pure acid needs to be added to make the solution 30% acid.

Number Problem 11 Solution

This is a new problem as asked by a anonymous visitor which read: "a number added to five times its reciprocal is -6. find the number "

Ok non-algebraically you can think of what numbers added together will make 6(of course they will be negative to make negative 6) ---  1 and 5  and 4 and 2 , and 6 and 0...  You can't have  a reciprocal of 0 and a reciprocal of 6 : 1/6 when multiplied by 5 makes 5/6ths and you know those added together :  -6 + - 5/6ths dont make -6.   Same thing goes for 2 and 3  --> 2 + 5/2 or  3 + 5/3 dont make 6.   1 and 5 however do work.   The reciprocal of 1 is 1 so 1 + 5/1 =6  and so does 5 + 5/5 = 6 (Of course negative again).   So there are 2 answers -1 and -5...

Algebraically...   -x + (1/-x)5 = -6  =

-x -5/x = -6  (multiply by -x)

x^2 + 5 = 6x

x^2 + 6x + 5 = 0

(x + 1) (x + 5 )    = 0 

So the answers are x = -1 and -5

Age Problem 6 Solution

This is the answer to the age algebra word problem that asked, "Bob is one third the age of his father. In 12 years he will be half the age of his father. How old is each now?"


Ok Bob being 1/3 the age of his father is the same thing as saying the father is 3 times as old as him.  It makes it easier to work with.  So let Bob be x and his father be 3x. And in 12 years bob will be half as old as his father.  So we have..


x + 12 =  (3x + 12) / 2

2x + 24 = 3x + 12

x  = 12


So Bob is 12 years old and his father is 3(12) or 36 years old now.  In 12 years Bob will be 12 + 12  or 24 years old and his father will be 36 + 12 or 48 years old which means Bob will be half as old so it checks out.

Age Problem 5 Solution

This is the algebra age word problem that read Pol is 10 years younger than greg. In 7 years, he will be 10 years more than one halfas old as greg. Find their age at present. help me solve it.

Let Greg be X and Pol be X - 10 as far as current ages.  In 7 years (thats x + 7) he will be 10 years more than 1/2 as old as greg (thats (x - 10 + 10)/2 or x/2).  So set them equal..

X  and x -10

x + 7  and  x - 10 + 7  28  and 25 



(x+7)/2 + 10 = x - 3

x + 17 = 2x - 6  ******** Correction    2 * 10 = 20 + 7 = 27 So it should be

x + 27 = 2x - 6

x =33  ---> Which is Greg's age so Pol's is x-10 or 33 -10 = 23 

So Pol is 23 and Greg is 33

Work Problem 6 Solution

This is the solution to Work Problem 6 which asked, "A mother can rake a yard in 90 minutes and her daughter can do it in 60 minutes.If the mother rakes for 15 minutes before her daughter joins her,how long will it take them to finish the work?"

Ok the mother can do 1/90th of her job in a minute  so if you have x/90  in 90 minutes 90/90 = 1 she completes her job same thing goes for the the daughter .. 1/60 of her job done in a minute with x/60 in 60 minutes 60/60 = 1 she completes her job.  Seems kind of silly that I'm pointing this out but this is how you solve the problem you just add their work together and set it equal to 1.  Their is one additional stipulation in the problem though and thats the mother starting 15 minutes before the daughter joins the mother otherwise it would just be
x/90 + x/60 = 1...  but the mother rakes for 15 minutes and completes 15/90 of her job so the way to write it out would be  x/90 + 15/90 + x/60 = 1.

multiply everything by 180 and get  2x + 30 + 3x = 180.

5x + 30 = 180
5x = 150
x = 30  So it would take 30 minutes if they worked together.   UPDATE:  "how long will it take them to finish the work?" <-- Sounds to me  like how long will they finish together  thus 30 minutes the answer.  But if you want to take this to mean how long did it take them to finish the total job then add the 15 minutes of initial work done by the mother which would be 30 + 15 = 45.

Plane Geometric Figure Problem 2 Solution

This is the Plane Geometric Figure Problem 2 Solution which asked "The length of the second side of a triangle is four less than three times the length of the first side. The length of the third side is one more than the length of the first side. If the perimeter of the triangle is 37 feet, what is the length of the first side?"

Since everything is relative to the 1st side make the 1st side x.

The 2nd side is 3x - 4 (4 less than 3 times the length of the 1st side)

and the 3rd side is x + 1 ( 1 more than the length of the 1st side) 

Now all the sides add up to 37 feet so set all this equal to 37.

x + 3x - 4  + x + 1 = 37

5x - 3 = 37

5x = 40 

x = 8

So the 1st side is 8 the 2nd side is 3(8) - 4 or 20 and the 3rd side is 9 .. 8 + 20 + 9 = 37

Finance Problem 2 Solution

This is the finance problem 2 solution which read: "A local furniture store is selling all $850 mattresses at 35% off. Alyssa is buying a mattress and has a coupon for an additional 10% off the sale price. What will Alyssa pay for her mattress? "

Ok you just have to take the discount off twice.   35% off of $850 would be $297.50  so 850 - 297.50 = $552.50 is the sale price.  Now she has a coupon for another 10% off that sales price.  So 10% of 552.50 is 55.25 and 552.50 - 55.25 = 497.25

So Alyssa will pay $497.25 for her mattress....

Coin Problem 6 Solution

This is the coin problem 6 solution which read "a stack of pennies and dimes has a total value of $2.31. how many dimes are in the stack if there are twice as many dimes as pennies?"

make $2.31 into cents so 231 cents .. a dime is worth 10x and penny is worth x.  So if you had 1 dime and 1 penny you would have 10(1) or 10 cents plus 1(1) or 1 cent which is 11 cents.  Were trying to have it add up to 231 cents though and there are twice as many dimes 2(10x) or 20x as pennies.  

So we have 20x + x = 231


21x = 231

x = 11  So there are 11 penny's or 11 cents and 22 dimes or 220 cents .. 220 cents plus 11 cents equals 231 cents or $2.31.

Number Problem 10 Solution

This is number problem 10 solution which read:

"Find 3 consecutive even integers such that 4 times the first is decreased by the second is 12 more than twice the third. " Ok I'm not understading the "is decreased by the" part so I assume it was a typo and you just meant "Find 3 consecutive even integers such that 4 times the first is 12 more than twice the third. "
 
ok 3 consecutive even integers would b x, x + 2, and x + 4

4 times the 1st would be just 4x and is equal to 12 more than twice the 3rd which would be (x + 4) * 2 + 12

which is...  4x = (x+4)*2 + 12

4x = 2x + 8 + 12

2x = 20

x = 10  So the answer would be 10, 12, 14   to check  4(10)  =  (14)*2 + 12

40 = 28 + 12

40 = 40

Work Problem 5 Solution

This is the solution to the work problem # 5 which read: Jim can fill a pool carrying buckets of water in 30 minutes. Sue can do the same job in 45 minutes. Tony can do the same job in 1 ½ hours. How quickly can all three fill the pool together?



Ok Tony takes 1 and 1/2 hours which is 90 minutes, and then we got jim at 30 minutes and Sue at 45 minutes. Convert everything to find out how much each can do in 1 minute. Tony does 1/90 th of his job in 1 minute Jim 1/30 th and Sue 1/45 th. So lets take just one individual persons job say Sue for example. If someone says Sue takes how long to finish her job if you know that she can do 1/45 of her job in a minute what would u say? 45 minutes. Or you can look at it as x/45 = 1 which means how many over 45 would = 1? The answer of course would be 45 again. So we set this problem up just like that only were ADDING the other peoples jobs together and setting it equal to 1.

So we got x/45 + x/90 + x/30 = 1 *Multiply both sides by 90 and get

2x + x + 3x = 90

6x = 90

x= 90/6 = 15

So it would take 15 minutes if all 3 worked together to finish the job.