$M is just sufficient to pay for the wages of one qualified teacher for x days or the wages of one relief teacher for y days. For how many days will $M be sufficient to pay for the wages of one qualified teacher and one relief teacher? Give your answer in terms of x and y.

Qualified teacher cost= M/x per day

Relief teacher cost = M/y per day
So,
= M(1/x + 1/y)
= xy/x+y

Ah, the math of budgets and teachers, such a delicate equation to solve. Now, let's see if we can find the answer with a sprinkle of humor.

If $M is just enough to pay for one qualified teacher for x days, then we can write it as:

$M = 1 qualified teacher × x days

And if $M is also just enough to pay for one relief teacher for y days, we can write:

$M = 1 relief teacher × y days

To calculate how long $M will be sufficient for both, we can add the two equations together:

$M + $M = (1 qualified teacher × x days) + (1 relief teacher × y days)

Now, let's simplify:

2$M = (1 qualified teacher + 1 relief teacher) × (x days + y days)

And there you have it! $M will be sufficient to pay for the wages of one qualified teacher and one relief teacher for x + y days. Boom, equation solved with a twist of humor!

To find the number of days $M will be sufficient to pay for the wages of one qualified teacher and one relief teacher, we need to determine the combined wages of the two teachers.

Let's assume the daily wage of the qualified teacher is $Wq and the daily wage of the relief teacher is $Wr.

For one qualified teacher, the total wages for x days would be:
Total wages for one qualified teacher = $Wq * x

For one relief teacher, the total wages for y days would be:
Total wages for one relief teacher = $Wr * y

To find the combined wages for one qualified teacher and one relief teacher, we can add the individual total wages:
Combined wages for one qualified and one relief teacher = ($Wq * x) + ($Wr * y)

Now, we need to determine how many days $M will be sufficient to pay for the combined wages. Let's assume this number of days is D.

The total wages for the combined teachers over D days would be:
Total wages for combined teachers = ($Wq * x) + ($Wr * y) * D

Since $M is just sufficient to pay for the wages, we can set the total wages equal to $M:
($Wq * x) + ($Wr * y) * D = $M

To find the number of days D, we can solve the equation for D:
($Wq * x) + ($Wr * y) * D = $M

Dividing both sides of the equation by ($Wq * x) + ($Wr * y), we get:
D = $M / (($Wq * x) + ($Wr * y))

Therefore, the number of days $M will be sufficient to pay for the wages of one qualified teacher and one relief teacher is given by:
D = $M / (($Wq * x) + ($Wr * y))

To find out how many days $M will be sufficient to pay for the wages of one qualified teacher and one relief teacher, we first need to determine the wages of each teacher per day.

Let's assume the wage of one qualified teacher per day is $Q, and the wage of one relief teacher per day is $R.

According to the given statement, $M is sufficient to pay for the wages of one qualified teacher for x days. Therefore, the wage of one qualified teacher per day is calculated as follows:
Wage of one qualified teacher per day = $M / x = $Q

Similarly, $M is also sufficient to pay for the wages of one relief teacher for y days. Therefore, the wage of one relief teacher per day is calculated as follows:
Wage of one relief teacher per day = $M / y = $R

Now, we need to determine the number of days $M will be sufficient to pay for the wages of one qualified teacher and one relief teacher. Let's assume this number of days is Z.

The combined daily wage of both teachers is calculated as:
Combined daily wage = Wage of one qualified teacher per day + Wage of one relief teacher per day
Combined daily wage = $Q + $R = $M / x + $M / y

To find the number of days $M will be sufficient, we divide the total amount of money $M by the combined daily wage:
Number of days = $M / Combined daily wage
Number of days = $M / ($M / x + $M / y)

Simplifying the expression, we get:
Number of days = (x * y * $M) / ($M * y + $M * x)
Number of days = (x * y) / (y + x)

So, the number of days $M will be sufficient to pay for the wages of one qualified teacher and one relief teacher is (x * y) / (y + x).

Okay... you've got $M, and you can afford one qualified teacher for x days, so a qualified teacher costs M/x dollars per day. Alternatively, you can afford a relief teacher for y days, so a relief teacher costs M/y dollars per day. So, if you're paying for one qualified teacher AND one relief teacher, you'll be spending (M/x + M/y) dollars per day, yes? That's M(1/x + 1/y) = M(x+y)/xy dollars per day. But you've only got $M to spend in total, so the number of days you'll have them both for must be $M divided by $M(x+y)/xy. The $Ms cancel out, leaving 1 / ((x+y)/xy) = xy / (x + y). That's my answer.

Now do a quick sanity check on that. Suppose a relief teacher costs almost nothing. Then you can afford one for a very long time, so y is extremely large. xy/(x+y) = x/(1+x/y), and as y increases without limit, this expression tends to x, which is the amount of time you can afford one qualified teacher - and that makes sense. Now suppose they cost the same. That would be like having two qualified teachers, the amount of time for which will be (x^2)/(2x) = x/2. That makes sense also - so I reckon we've got it right.