math word problem question

Question

the second of three numbers is six less than twice the first. the third number is four times the first. if the sum of the first and the third is 9 more than four times the second, find the number

Answer 1

X is the first, as know nothing about it.

The second is 2x-6

The third is 4x

The sum of the first and third is 9 more than

4 times the second...

X + 4x = 4(2x-6)+9

X+4x=8x-24+9

5x=8x-15

0=3x-15

15=3x

X=5

The first number is 5.

The second number is 2*5-6=4

The third number is 20

Answer 2

Word problems such as these are fun in that one gets to sleuth-out everything and sort of having fun while arriving at the answer. Now, to start-out, one needs to carefully craft the words being given and take it one step at a time.

Let us, first, define the variables which will represent the numbers:

let x = the first number

let y = the second number

let z = the third number

The problem starts out with the statement:

"The second of three numbers is six less than twice the first"

Now, let us interpret this sentence one statement at a time:

1. The second of three numbers:
2. This is referring to the number which we labelled as "y"
3. six less than twice the first:
4. twice the first means the first number, "x", is multiplied by 2
5. six less means 6 is subtracted from the product
6. obtaining ---> 2x - 6
7. the two statements are joined by the word "is" which means "equals to", therefore, we now have:
8. y = 2x - 6

The second statement reads:

"The third number is four times the first."

Interpreting this sentence, which is made-up of a single statement:

1. "The third number"
2. This refers to the one which we labelled as "z"
3. "four times the first"
4. meaning 4 times the first number, "x", giving 4x
5. The two statements are joined by the word "is" which means "equals to," therefore, we now have:
6. z = 4x

The last statement:

"If the sum of the first and the third is nine more than four times the second"

let us break down this sentence, which is made-up of two statements:

1. If the sum of the first and the third
2. "the sum of the first and the third"
3. meaning x + z
4. nine more than four times the second
5. "four times the second" means 4 multiplied by "y" therefore 4y
6. "nine more" means 9 added to 4y
7. therefore we now have 4y + 9
8. the is a word joining these two statements, this is the word "is" which means "equals"
9. therefore we have
10. x + z = 4y + 9

Now, we are dealing with this final equation:

x + z = 4y + 9

From the first sentence, it was determined that the "second of three numbers is six less than twice the first" and that equation is:

y = 2x - 6

Substituting this value of "y" into the equation above it yields:

1. x + z = 4y + 9
2. x + z = 4 (2x - 6) + 9
3. x + z = 8x - 24 + 9
4. x + z = 8x - 15

Our resulting equation is:

x + z = 8x - 15

From the second statement, ""The third number is four times the first," we determined that:

z = 4x

Taking this value of "z" and substituting it into the equation above it yields:

1. x + z = 8x - 15
2. x + 4x = 8x - 15
3. 5x = 8x - 15
4. -3x = -15
5. x = 5

Therefore, the number for which we are looking is:

x = 5

The value of the second number is therefore:

y = 2x - 6

y = 2 (5) - 6

y = 10 - 6

y = 4

The value of the third number is therefore:

z = 4x

z = 4 ( 5 )

z = 20