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

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

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....

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....

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

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

Number Problem 9 Solution

This is the solution to the algebra number word problem which was asked by a anonymous person in the ask question section which read, "One number is 5 more than another.their sum is 53.what are the two numbers?"


So we have one number is x and another is 5 more than it which is x + 5. they equal 53.

So we have x + x + 5 = 53

2x + 5 = 53

2x = 48

x = 48/2 = 24 for one number and 24 + 5 or 29 for the other.

So check 24 + 29 = 53

53 = 53

Number Problem 8 Solution

This is the solution to the Algebra Number word problem # 8 which read "The ratio of 16 more than a number to 12 less than that number is 1 to 3. What is the number?"



Ok a number is x - the unknown

The problem says that ration of 16 more than a number --> which is x + 16 to 12 less than that number(the number is still x) or x - 12 is 1 to 3

So we have (x + 16)/(x -12) = 1/3
Cross multiply and you get 3*(x + 16) = 1*(x - 12)

3x + 48 = x - 12

2x = - 60

x = -30

Check:
(-30 + 16)/(-30 - 12) =
-14/-42 = 14/ 42 = 1/3

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

Tags: Number Problem Solutions, algebra word problem solutions

Number Problem 6 Solution

This is the solution to the number problem # 6.

In the following Algebra number word problem we have a 3 digit number which has a 100's digit 3 more than the 1's digit and its 10's digit is twice the 100's digit. When all 3 digits are added together, it is equal to 9. What is the number?

If you make the 1's digit x then you have:

100's digit = x + 3 and the

10's digit = 2(x + 3) or 2x + 6

Now when all these digits are added together you get 9 so...

x + x + 3 + 2x + 6 = 9

4x = 0

x = 0

so if x = 0(the 1's digit) then the 100's digit would be 0 +3 or just 3 and the 10's digit would be 0 + 6 or 6.

so the answer is 360

Technorati Tags: Number Problem Solutions, Algebra Word Problem Solutions

Number Problem 5 Solution

This one's a lot easier to work algebraically. You could get it by trial an error or course, but its easy if you think of one number being just x and the other number being (40-x)

so for example if the number was 10 then the other number would be (40 - 10) or 30

*this is pretty close if you check 5(10) = 50 2(30) = 60 which is a 10 difference and not a 3 through trial and error you could get the right answer just switching out a new number

Algebraically:

So 2 times the 1st number lets use (40-x) since multiply by 2 with this is easier than by 5

2 (40 - x) is "equal" to

3 more than 5 times another number (x)

or 5x + 3

altogether we have 2(40-x) = 5x + 3

80 - 2x = 5x + 3
77 = 7x
x = 11

so 1 number is 11 and the other number is (40 - 11) or 29

check it--- 2(29) = 58 5(11) = 55 and indeed they are 3 apart

Number Problem 4 Solution

You might be able to figure this out in your head since it doesn't matter if the number was 900, 9 million etc if its a situation that add up to 9 and 1 number is twice as big that you're adding. So what numbers 1 through the number 9 could you add together to get 9? 1 and 8 , 2 and 7, 3 and 6, and 4 and 5. Now which one of these is twice as big as the 1st number? The 6 is twice as big as the 3. But the problem asked for what 2 numbers added up to 900. So just add 2 zeros onto the 6 and 3. 300 and 600 are the answers.

Algebraically,

x = 1st number
2x = 2nd number *2x means the same thing as 2 times x or 2 X x
900 is the number were looking for so..

x + 2x = 900
3x = 900
x = 900/3
x = 300
substituting back in x is 300 and 2x is 2(300) or 2 X 300 which is 600.

Number Problem 3 Solution

Odd numbers are 5, 11, 17, 19, 21, etc and consecutive means one right after the other as in 3,4,5,6, etc . So 3 consecutive odd numbers means 45,47,49 for example.

You want 3 consecutive odd that add up to 51 so the 1st number is x, the next number is x +2, and the last number is (x + 2) + 2 or x + 4. So,

x + x + 2 + x + 4 = 51 which equals

3x + 6 = 51 =

3x = 45

x = 45/3 = 15

So substituting back in for x you have 15 + (15 + 2) + (15 +4) = 51

15 + 17 + 19 does = 51 which is the answer ...

Number Problem 2 Solution

Even numbers are 2, 6, 12, 18, etc- consecutive means one right after the other 1,2,3,4-- so consecutive even means 6,8,10,12, etc ...

So, the 1st number would be x, the 2nd number x + 2, and the 3rd number would be (x + 2) + 2 or x + 4 . The problem states when these 3 numbers are added together they equal 138.

We then have x + x + 2 + x + 4 = 138 which makes
3x + 6 = 138 which makes
3x = 132
which makes x = 132/3
which is x = 44

Substituting back in we have 44 + 44 + 2 + 44 + 4 = 138 and that equals
44 + 46 + 48 = 138 so 138 = 138

The 3 consecutive even numbers when added together that equal 138 are 44, 46, 48

Number Problem 1 Solution

Consecutive means 1 right after the other as in 3,4,5,6,7.... So in this problem the 1st number is x the next number is x + 1 and the last number is (x+1) +1 which is x + 2.

So we have x + x + 1 + x + 2 = 48
3x = 45 x = 15 ..... substituting back in the problem
15 + (15 + 1) + (15 + 2) = 48 = 15 + 16 + 17 = 48 ...
so the 3 consecutive numbers are 15,16,17.