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Lesson 4: Equations word problems

# Sums of consecutive integers

When solving a math problems involving consecutive integers, we can use algebra to set up an equation to solve for the smallest of the four integers. Created by Sal Khan.

## Want to join the conversation?

• a man drove 48 more miles an hour than the number of hours he drove. if he drove 208 miles in all, how many hours did he drive?what was his rate?
• 48/1=208/?..... Approximately 4 hours and 20 minutes, I guess. His rate was 48 miles per hour and 4.33333 hours per 208 miles.
• Is it not more simple to divide 136 by 4 and take the two odd pairs on either side of the result?
• It could be, but he did it via the counting method.
• I used a different method... The results are the same, but the method itself is a bit more complex. We can assume that an even integer comes in the form of 2n and hence an odd number is 2n+1. Using this method, the four integers are:
2n+1
2n+3
2n+5
2n+7
Their sum is 8n+16=136.
8n=120
n=15
And then the first number is 2n+1=2*15+1=30+1=31
the second number: 2n+3=31+2=33
and so on.

The reason I'm putting this in the questions section is because I was wondering if there is a moment when this method is better than the one presented... Or is it just a more complicated way of doing something simple?
• In math there are often many different approaches to solving a given problem. Your approach is fine and if it seems more intuitive for you, go for it. Others may or may not find your way more intuitive than what is demonstrated in the video, so thanks for sharing your solution!
• how do you know what the next integer is going to be like if its x+(x+1)+(x+2) or if its x+(x+2)+(x+4)
• If it's [consecutive integers], it's going to be x+(x+1)+(x+2), since the common difference between consecutive integers is 1.
If it's [consecutive odd/even integers], it's going to be x+(x+2)+(x+4), since the common difference between odd or even integers is 2.
• When Khan requested that I pause and attempt to solve the problem on my own, here's what I did:

1) Divided 136 by 4 (ie took the average of the four consecutive odd ints)
2) Result: 34
3) Formed a mental number line around 34 like so: 31-32-33-34-35-36-37
4) Selected the two nearest odd ints to the left of 34 and the two nearest odd ints to the right on my mental number line.
5) added them up to confirm they equal 136

Is there anything wrong with this approach? Are their any scenarios where this approach doesn't work for INTEGERS? (obv. fractions are another matter and Sal's algebraic approach is preferred then).
• Yes, your method will always work for integers -- and if you can picture the number line in your head clearly enough then often it will be quicker, too. The algebraic method is very useful, though, with situations where it's hard to picture the problem in your head.
• this video makes no sense
• Watching it twice and reviewing previous lessons can help with understanding. :)
• So, when doing consecutive evens we skip count evens x + x+2, x+4, x+6, ....etc? Then odds we use x, x+1, x+2, x+3? I noticed the set up was different depending on what the problem asked (evens/odds)? How can we tell the difference?
• no, odds are still x, x+2, x+4 etc. also because odd numbers are two apart from each other just like even numbers are. if x =3, then x+1 = 4, but that is not odd. The second pattern you used is for consecutive numbers only.
• In these problems, you don't know the value of the integers. Since consecutive integers are 1 unit higher from each other, you leverage that pattern to define variable representations for the unknown integers:
x = 1st integer
x+1 = 2nd integer
x+2 = 3rd integer
etc.

If the problem deals with consecutive odd or even integers, these increment by 2. So the pattern becomes:
x = 1st odd/even integer
x+2 = 2nd odd/even integer
x+4 = 3rd odd/even integer
etc.

You can extend the pattern to have as many integers as you need for the problem you are doing. Sal needed to find 4 odd consecutive integers, so he extended the pattern out to x+6. Then, apply the math described in the word problem to set up an equation. Sal's problem, asked for the sum of the numbers, so his equation became adding the 4 unknown consecutive odd integers.

Hope this helps.
• The sum of three consecutive integers is 53 more than the least of the integers. How would I do this?