In a set of five consecutive integers, the smallest integer is more than 2/3 the largest. What is the smallest possible value of
2 answers:
The answer is: "55".
________________
"55" is the smallest possible value of the sum of the five integers.
________________________________________
We are given five (5) consecutive integers.
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Let us represent the first integer as: "x";
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The second integer as: "x + 1" ;
___________________________________
The third integer as: "x + 2" ;
___________________________________
The fourth integer as: "x + 3" ;
___________________________________
The fifth integer as: "x + 4" ;
___________________________________
We are given: The smallest integer is MORE THAN (⅔ of the largest integer.
____________________________________
The smallest integer, "x" is greater than: "(⅔) (x + 4)"
________________________________________________
Note the distributive property of multiplication:
__________________________________
→ a*(b + c) = ab + ac ;
__________________________________
→ As such; (⅔) (x + 4) = [(⅔)*(x)] + [⅔)*(4);
________________________________________
→ (⅔) (x + 4) = [(⅔)*(x)] + [⅔)*(
)
______________________________________________
{Note: [⅔)*()=
=
};
_______________________________________________
→ Rewrite the equation: → ⅔ of the largest integer =
______________________
→ (⅔) *(x + 4) = (⅔)x + ;
_______________________________
The smallest, "x", is GREATER than: (⅔)x + ;
________________________________
→ Write as: x > (⅔)x +
______________________________________
→ Multiply the ENTIRE "inequality" (BOTH SIDES) by "3", to get rid of the "fractions":
_______________________________
3*{ x > (⅔)x + } ; to get:
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→ 3x > 2x + 8 ; → Now, subtract "2x" from EACH SIDE of the inequality;
_________________________
→ 3x − 2x > 2x + 8 − 2x ;
______________________
→ to get: → x > 8
____________________
Now, we want to to know the "smallest possible value of the sum five integers".
___________________________________
→That is, the smallest possible value of the sum of :
___________________________________________
→x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 5x + 10.
__________________
So, we want to know the smallest value of "(5x + 10)" ; x > 8,
_______________________________________________
→ Let us solve for "5x + 10" ; when x = 10; by substituting "8" for "x"
________________
→ 5x + 10 = (5*8) + 10 = 40 + 10 = 50
___________________________________
So, if "x > 8"; then "x" must be greater than "8";
_____________________________
When "x = 8", the sum of the 5 (five) integers in our problem = 50.
___________________________
Since "x" in the first of the consecutive integers in the problem; we know that "x > 8"; then the smallest possible value for "x" would be "9"; since "x" has to be an integer.
______________________________
So, we know that:
The smallest possible value of sum of the value integers =
5x + 10; when x = 9;
_________________
→ So, we plug "9" for the value of "x" and solve:
______________________________
→ 5(9) + 10 = 45 + 10 = 55 ; → which is our answer.
_____________________________________
Let us check our answer:
_______________________
→ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = ? 55?
(when "x = 9" ?)?? ;
___________________________________________________
→ 9 + (9 + 1) + (9 + 2) + (9 + 3) + (9 + 4) = ? 55? ;
______________________________________
→ 9 + 10 + 11 + 12 + 13 = ? 55 ?? ; Yes!.
__________________________________________
Also, is the answer: 55 , Reasonable? Yes, since it is an integer, and 5 consecutive integers added together would "add up" to an integer value.
__________________________________________
Hope this answer and lengthy explanation is helpful.
Best of luck!
_______________________
________________
"55" is the smallest possible value of the sum of the five integers.
________________________________________
We are given five (5) consecutive integers.
___________________________________
Let us represent the first integer as: "x";
___________________________________
The second integer as: "x + 1" ;
___________________________________
The third integer as: "x + 2" ;
___________________________________
The fourth integer as: "x + 3" ;
___________________________________
The fifth integer as: "x + 4" ;
___________________________________
We are given: The smallest integer is MORE THAN (⅔ of the largest integer.
____________________________________
The smallest integer, "x" is greater than: "(⅔) (x + 4)"
________________________________________________
Note the distributive property of multiplication:
__________________________________
→ a*(b + c) = ab + ac ;
__________________________________
→ As such; (⅔) (x + 4) = [(⅔)*(x)] + [⅔)*(4);
________________________________________
→ (⅔) (x + 4) = [(⅔)*(x)] + [⅔)*(
______________________________________________
{Note: [⅔)*(
_______________________________________________
→ Rewrite the equation: → ⅔ of the largest integer =
______________________
→ (⅔) *(x + 4) = (⅔)x +
_______________________________
The smallest, "x", is GREATER than: (⅔)x +
________________________________
→ Write as: x > (⅔)x +
______________________________________
→ Multiply the ENTIRE "inequality" (BOTH SIDES) by "3", to get rid of the "fractions":
_______________________________
3*{ x > (⅔)x +
________________________________________
→ 3x > 2x + 8 ; → Now, subtract "2x" from EACH SIDE of the inequality;
_________________________
→ 3x − 2x > 2x + 8 − 2x ;
______________________
→ to get: → x > 8
____________________
Now, we want to to know the "smallest possible value of the sum five integers".
___________________________________
→That is, the smallest possible value of the sum of :
___________________________________________
→x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 5x + 10.
__________________
So, we want to know the smallest value of "(5x + 10)" ; x > 8,
_______________________________________________
→ Let us solve for "5x + 10" ; when x = 10; by substituting "8" for "x"
________________
→ 5x + 10 = (5*8) + 10 = 40 + 10 = 50
___________________________________
So, if "x > 8"; then "x" must be greater than "8";
_____________________________
When "x = 8", the sum of the 5 (five) integers in our problem = 50.
___________________________
Since "x" in the first of the consecutive integers in the problem; we know that "x > 8"; then the smallest possible value for "x" would be "9"; since "x" has to be an integer.
______________________________
So, we know that:
The smallest possible value of sum of the value integers =
5x + 10; when x = 9;
_________________
→ So, we plug "9" for the value of "x" and solve:
______________________________
→ 5(9) + 10 = 45 + 10 = 55 ; → which is our answer.
_____________________________________
Let us check our answer:
_______________________
→ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = ? 55?
(when "x = 9" ?)?? ;
___________________________________________________
→ 9 + (9 + 1) + (9 + 2) + (9 + 3) + (9 + 4) = ? 55? ;
______________________________________
→ 9 + 10 + 11 + 12 + 13 = ? 55 ?? ; Yes!.
__________________________________________
Also, is the answer: 55 , Reasonable? Yes, since it is an integer, and 5 consecutive integers added together would "add up" to an integer value.
__________________________________________
Hope this answer and lengthy explanation is helpful.
Best of luck!
_______________________
You might be interested in
Answer:
6√3
Explanation:
Before we begin, remember the following:
Now, for the given:
75 can be written as 25*3
This means that:
Applying the above concept, we can find that:
Now, we know that:
√25 = 5 (we ignored the negative value)
This means that:
√75 = 5√3
Finally, we can compute the needed sum as follows:
√75 + √3 = 5√3 + √3 = 6√3
Hope this helps :)
6√3
Explanation:
Before we begin, remember the following:
Now, for the given:
75 can be written as 25*3
This means that:
Applying the above concept, we can find that:
Now, we know that:
√25 = 5 (we ignored the negative value)
This means that:
√75 = 5√3
Finally, we can compute the needed sum as follows:
√75 + √3 = 5√3 + √3 = 6√3
Hope this helps :)