SOLUTION: James spent $6780 on some watches and clocks. The amount spent on watches was $2820 more than the amount spent on the clocks. He bought 3/5 as many clocks as watches. Each clock co

Question 1199194: James spent $6780 on some watches and clocks. The amount spent on watches was $2820 more than the amount spent on the clocks. He bought 3/5 as many clocks as watches. Each clock cost $25 less than each watch. What was the total number of watches and clocks bought by James?

Found 3 solutions by math_tutor2020, MathTherapy, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Given facts:
James spent $6780 on some watches and clocks.
The amount spent on watches was $2820 more than the amount spent on the clocks.
He bought 3/5 as many clocks as watches.
Each clock cost $25 less than each watch.
Let
w = number of watches

Since 3/5 = 0.6, we can use fact 3 to say:
0.6w = number of clocks

Now let
x = cost of 1 watch
which would mean
x-25 = cost of 1 clock
due to fact 4

1 watch = x dollars
w number of watches = wx dollars
1 clock = (x-25) dollars
0.6w number of clocks = 0.6w(x-25) dollars

Add these subtotals to get $6780 spent on everything
Refer to fact 1 above.
watches + clocks = 6780 total
wx + 0.6w(x-25) = 6780
wx + 0.6wx - 0.6w*25 = 6780
1.6wx - 15w = 6780
Pause here for now. We'll come back to this later.

Fact 2 can be rephrased as:
"The amount spent on clocks was $2820 less than the amount spent on the watches"
which means:
wx = amount spent on watches
wx-2820 = amount spent on clocks

Those two subtotals also add to 6780, due to fact 1.
wx+(wx-2820) = 6780
2wx-2820 = 6780
2wx = 6780+2820
2wx = 9600
wx = 9600/2
wx = 4800
Buying w number of watches, at x dollars a piece, cost $4800.

wx-2820 = 4800-2820 = 1980
and the clocks cost $1980.

We'll now return to the previous equation we paused at.
Plug in wx = 4800
So,
1.6wx - 15w = 6780
1.6(wx) - 15w = 6780
1.6*(4800) - 15w = 6780
7680 - 15w = 6780
-15w = 6780-7680
-15w = -900
w = -900/(-15)
w = 60
James bought 60 watches.
Each watch costs x = (wx)/w = 4800/60 = 80 dollars.

0.6w = number of clocks
0.6w = 0.6*60
0.6w = 36
and he also bought 36 clocks.
Each clock costs x-25 = 80-25 = 55 dollars.
He spent 36*55 = 1980 dollars on clocks as mentioned earlier.

The two subtotals should add to 6780 dollars.
watches + clocks = 4800+1980 = 6780
This helps confirm the answers.

Further confirmation is to note that
4800-1980 = 2820
which matches up with fact 2

There's probably a much more faster/efficient way to solve this.
Feel free to explore other options.

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Answer:
60 watches + 36 clocks
96 total items.

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

James spent $6780 on some watches and clocks. The amount spent on watches was $2820 more than the amount spent on the clocks. He bought 3/5 as many clocks as watches. Each clock cost $25 less than each watch. What was the total number of watches and clocks bought by James?

Let amount spent on watches be W Since a total of $6,780 was spent on watches and clocks, amount spent on clocks = 6,780 - W Since amount spent on watches was $2,820 more than what was spent on clocks, then amount spent on clocks = W - 2,820 We then get: 6,780 - W = W - 2,820 - W - W = - 2,820 - 6,780 - 2W = - 9,600 Amount spent on watches =, or Since $4,800 was spent on watches, amount spent on clocks = 6,780 - 4,800 = $1,980 Now, let number of watches purchased be W, and cost of each watch, C Then number of clocks purchased = , and cost of each clock: C - 25 With total spent on watches being 4,800, we get: WC = 4,800 ----- eq (i) Also, with total spent on clocks being $1,980, we get: 3WC - 75W = 9,900 ----- Multiplying by LCD, 5 ----- eq (ii) 3(4,800) - 75W = 9,900 ----- Substituting 4,800 for WC in eq (ii) 14,400 - 75W = 9,900 - 75W = 9,900 - 14,400 - 75W = - 4,500 Numner of watches purchased, or Number of clocks purchased:

Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
James spent $6780 on some watches and clocks.
The amount spent on watches was $2820 more than the amount spent on the clocks.
He bought 3/5 as many clocks as watches.
Each clock cost $25 less than each watch.
What was the total number of watches and clocks bought by James?
~~~~~~~~~~~~~~~~

        The solution consists of two parts (two steps) and uses
        two clear simple ideas, accompanying with short calculations.

                                First step

Let X be the amount of money spent on watches, Y be the amount of money spent on clocks. From the condition, we have two equations X + Y = 6780 (total dollars) X - Y = 2820 (X is $2820 more than Y) To find X, add the equations. You will get 2X = 6780+2820 = 9600, X = 9600/2 = 4800 dollars. To find Y, subtract second equation from the first one. You will get 2Y = 6780-2820 = 3960, Y = 3960/2 = 1980 dollars.

First step is complete. We just found that the amount spent on watches was $4800; the amount spent on clocks was $1980.

                                Second step

Let W be the number of watches bought by James. Then the number of clocks was = 0.6W. Each watch price was dollars; each clock price was = . Next, the difference of prices is 25 dollars. It gives this "price" equation - = 25. Simplify and find W = 25 W = = 60. ANSWER. 60 watches and = 36 clocks. The total number of items was 60+36 = 96.

Solved.

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The meaning of this problem is not to make tons of calculations.
The meaning is to find a right idea of solution.

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