5 math problems

1.

The question is...

$8,880  is  6%  of what number?               Let's call the number  " p " for selling price.

$8,880  =  0.06 * p                                      Divide both sides by 0.06

$8,880 / 0.06  =  p

$148,000  =  p

A class average mark in an exam is 70. The average of students who scored below 60 is 50. The average of students who scored 60 or more is 75. If the total number of students in this class is 20, how many students scored below 

Note that if the class average is 70, then the total number of points scored is  70 * 20  = 1400

Call the number of students who scored below 60, x .....So....their total number of points they scored is just their average times their number =  50x

Call the number of students who scored above 60, 20-x .....So....their total number of points they scored is just their average times their number =  75 (20 - x)

So.....we have this equation :

50x + 75 (20 - x)  = 1400       simplify

50x + 1500 - 75x  = 1400      subtract 1500 from both sides

-25x = - 100      divide both sides by -25

x = 4  ..so.....4 scored below 60

For what value of x will the function f(x) = -3(x - 10)(x - 4) have a maximum value? Find the maximum value.

We could do some fancy calculations here, but note that this is an "upside-down" parabola that has zeroes at   x = 4  and at x = 10

And, because of symmetry, the max value will occur halfway between these, i.e., at x = 7

And the max value will be  -3(x - 10)(x - 4) =  -3(7 - 10) (7 - 4)  =  -3 * -3 * 3  =  27

Four children have small toys. The first child has 1/10 of the toys, the second child has 12 more toys than the first, the third child has one more toy of what the first child has and the fourth child has double the third child. How many toys are there? 

Let's call the total number of toys  " t " .

first child's toys  =  t/10

second child's toys  =  12 + first child's toys  =  12 + t/10

third child's toys  =  1 + first child's toys  =  1 + t/10

fourth child's toys  =  2 * third child's toys  =  2 * (1 + t/10)  =  2 + 2t/10

So.....

t  =  first child's toys + second child's toys + third child's toys + fourth child's toys

t  =  ( t/10 ) + ( 12 + t/10 ) + ( 1 + t/10 ) + ( 2 + 2t/10 )

t  =  t/10 + 12 + t/10 + 1 + t/10 + 2 + 2t/10                     Combine like terms

t  =  5t/10 + 15

t  =  t/2 + 15                Subtract  t/2  from both sides.

t - t/2  =  15

t/2  =  15                     Multiply both sides by  2  .

t =  30

In a shop, the cost of 4 shirts, 4 pairs of trousers and 2 hats is $560. The cost of 9 shirts, 9 pairs of trousers and 6 hats is $1,290. What is the total cost of 1 shirt, 1 pair of trousers and 1 hat? 

It might be difficult to solve this outright with a system of equations because we might have more variables than equations.....let's see if we might arrive at some logical conclusion...

Note that  if I divided the second thing by 3, we would have that

3 shirts, 3 pairs of trousers and 2 hats is $430

And taking the first we have that 4 shirts, 4 pairs of trousers and 2 hats is $560

Then. since the cost of the hats is the same, it must be that   1 shirt and 1 pair of trousers  cost

 $560 - $430  = $130

So....9 sets of  each of them must cost  9 * $130 = $1170

So....the cost of a hat =  [ 1290 - 1170] / 6  = 120/6  = $20

So....the total cost of 1 shirt, 1 pair of trousers and 1 hat =  $130 + $20 = $150

Check  

4(130) + 2(20)  = $560

9(130) + 6(20) = $1290