If the sum of 7 consecutive numbers is 140, which number is the smallest?

Understand the Problem

The question is asking for the smallest number in a set of 7 consecutive integers whose sum equals 140. To find the smallest number, we can denote the smallest number as 'x'. Therefore, the consecutive numbers can be represented as x, x+1, x+2, x+3, x+4, x+5, x+6. We can set up the equation x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) = 140 and solve for x.

Answer

$17$

Steps to Solve

Set up the equation for the sum of the integers
We need to sum the 7 consecutive integers represented as $x, x+1, x+2, x+3, x+4, x+5, x+6$. This can be expressed mathematically as: $$ x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) = 140 $$
Simplify the left side of the equation
Combine like terms on the left side: $$ 7x + (0 + 1 + 2 + 3 + 4 + 5 + 6) = 140 $$ The sum of the numbers $0 + 1 + 2 + 3 + 4 + 5 + 6$ equals $21$, so we rewrite the equation as: $$ 7x + 21 = 140 $$
Isolate the variable
Subtract $21$ from both sides to isolate the term with $x$: $$ 7x = 140 - 21 $$ This simplifies to: $$ 7x = 119 $$
Solve for x
Now, divide both sides by $7$: $$ x = \frac{119}{7} $$ This simplifies to: $$ x = 17 $$
Answer verification
Finally, we can check our answer by finding the consecutive integers: $17, 18, 19, 20, 21, 22, 23$. We add them up to ensure they equal $140$: $$ 17 + 18 + 19 + 20 + 21 + 22 + 23 = 140 $$ This confirms our solution.

The smallest number in the set of 7 consecutive integers is $17$.

More Information

This problem demonstrates the properties of consecutive integers and how to set up and solve a simple algebraic equation. The sum of a series of numbers can help illustrate the concept of averages and number series.

Tips

Failing to combine like terms correctly in the equation can lead to an incorrect final result.
Not checking the solution by summing the consecutive integers to ensure they match the original total.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!