An older person is 5 years older than six times the age of a younger person. The sum of their ages is 40. Find their ages

Question

i just need assistance with this problem i am stuck on it.

Answer 1

Hi Marry, this is a situation where putting the problem into equation form will help a lot!  First things first, since we have two ages we're trying to find, we'll need two variables, x and y.  Let's assign the older person the variable x, and the younger person the variable y.  Also, because we need to find two variables, we'll need to have two equations.  Don't let this scare you though, it's actually much simpler than it seems at first glance!

 

So, let's get down to it.  Looking back over the problem, we can find one of our equations right off the bat: "The sum of their ages is 40."  This translates very easily into an equation:

 

x + y = 40

 

One equation down, one to go!  Now, let's put that first equation in our back pocket for the time being and work on deciphering that first sentence.  "An older person is 5 years older than 6 times the age of a younger person."  That's a handful, but there are a couple of tricks that can help you decode sentences like this.  First of all, whenever you see the word "is" in a word problem, more than likely you can turn that straight into an equals sign.  In this case, that gives us one full half of the equation without having to do any work, because we've already assigned the older person the variable x.  So our first half of this equation is provided for us:

 

x =

 

The rest of the sentence will turn into the other half of that equation, centering around the younger person, who we've assigned the variable y.  So, what we're working with is: "5 years older than 6 times the age of the younger person".  By simply plugging in the variable we have already assigned to the younger person, things start to get easier: "5 years older than 6 times y."  One step further, since we know that the word "times" means "multiplied by", and we get: "5 years older than 6y."  Now, the final step, since "5 years older" can be translated to "plus 5 years", we get: "5 + 6y."  Plug that into the second half of our second equation, and we get:

 

x = 5 + 6y

 

Now we have our two equations and can start to solve this problem!  Remember, our two equations are:

 

x + y = 40

 

and

 

x = 5 + 6y

 

In order to solve for both x and y, we will need to do two sets of substitution: one to find x, one to find y.  We will be substituting one equation into another, but don't worry, it's not as bad as it sounds.  Let's start by finding y.  The way we do this is, we substitute x in one equation for its equivalent in y terms in the other, so that the resulting equation only has y variables.  If that's confusing, let's break it down.  We already know that in order to solve for a variable, we have to get it by itself on one side of the equation.  One of our equations, x = 5 + 6y, already has that.  x is by itself in that equation.  So let's get it by itself in the other one:

 

x + y = 40

 

In this equation, y is on the same side as x.  We want to get y on the other side so that x can be alone.  We accomplish this by doing the opposite of whatever y is currently doing.  So, in this equation, y is being added to x.  In order to get x alone then, we need to subtract y.  Remember that whatever you do to one side of an equation you have to do to the other!  So:

 

x + y – y = 40 – y

 

Which leaves us with:

 

x = 40 – y

 

Now both of our equations have x by itself:

 

x = 5 + 6y

 

and 

 

x = 40 – y

 

So now we can start to solve for y.  The first step is substituting the equivalent of x in one equation for the x variable in the other.  What this looks like is simply switching one equation's x side for the other equation's y side, or:

 

40 – y = 5 + 6y

 

Now all we do is solve for y.  We do this, again, by getting both y variables on the same side, and then removing all the other numbers from that side so that y is alone:

 

40 – y + y = 5 + 6y +y

 

Which gives us:

 

40 = 5 + 7y

 

Then:

 

40 – 5 = 5 + 7y – 5

 

Which gives us:

 

35 = 7y

 

Then:

 

35 = 7y

 7      7

 

Which gives us:

 

5 = y

 

Now we know the age of the younger person: 5.  With this, we can easily solve for x by going back to our original two equations, x + y = 40 and x = 5 + 6y.  It doesn't matter which one we choose; all we have to do is replace y with its value, which we now know to be 5.  So:

 

x + 5 = 40

 

And solve:

 

x + 5 – 5 = 40 – 5

 

Which leaves us with:

 

x = 35

 

Now we know both ages!  The older person, x, is 35.  The younger person, y, is 5.  That's the solution!  But it's always a good idea to check your work, so let's go through that process as well.  We do that by solving both of our two equations by replacing both x and y with their respective numerical values and making sure it's equal.  Our first equation looks like this:

 

x + y = 40

 

With the numbers substituted in, it looks like this:

 

35 + 5 = 40

 

And solved:

 

40 = 40

 

So far, so good.  Now on to our second equation:

 

x = 5 + 6y

 

Substituted, it looks like this:

 

35 = 5 + (6 * 5)

 

Notice that I've put 6 * 5 into parenthesis in order to avoid any confusion.  Now let's go through the steps of solving this. First we do our parenthesis:

 

35 = 5 + 30

 

And solved:

 

35 = 35

 

We're in the clear!  Everything matches up.  x is definitely 35, and y is definitely 5.  There's your answer!  I hope this helped.  Please let me know if you have any questions, or if you're interested in a tutor!

 

Answer 2

First, identify the quantities involved:

 

let s be the age of the older ("s" for "senior")

let y be the age of the younger

 

Next, translate the words into equations that relate these quantities:

 

First, the formula for the relationship

translate "an older person is 5 years older than...," into "s = 5+ "

translate six times the age of a younger person" into "6y"

string them together "s = 5 + 6y"

 

Second, the formula for the sum

translate "the sum of their ages is 40" into "s + y = 40"

 

Now you have two equations in two unknowns, and you know how to solve it.

s = 5 + 6y

s + y = 40