Question 204021
In solving a word problem you often use one or more variables and then one or more equations to solve it. Generally the number of variables dictates the number of equations. So it is preferable to use as few variables as possible.<br>
You can reduce the number of variables you use if you decide what one variable will represent and then try to express as many of the other unknowns in the problem using expressions involving that variable. And creating these expressions is usually easier if your variable represents the lowest (or near to the lowest) unknown in the problem.<br>
In your problem, I hope it is clear that the number of half dollars would be the lowest number in the problem. (Please read and think about the problem until this becomes clear.) So we will use our variable, x, to represent the number of half dollars. Now let's see how many of the other unknowns we can express in terms of x:
Let x = the number of half dollars.
Since she asks for "twice as many quarters as half dollars":
The number of quarters = 2*x = 2x
Since she asks for "twice as many dimes as quarters":
The number of dimes = 2*(2x) = 4x
Since she asks for "three times as many nickels as dimes":
The number of nickels = 3*(4x) = 12x
We have succesfully expressed the numbers of all the different coins in terms of a single variable! We only need one equation, then, to solve the problem.
The equation will state that the sum of the values of these coins adds up to $10. Therefore we will need expressions for the values of the coins.<br>
After some thought it we should be able to come up with the correct expressions.
Since 1 half dollar is worth $0.50 and 2 half dollars is worth $1.00 and 3 half dollars is worth $1.50, we should be able to figure out that the value of the half dollars is $0.50 times the number of half dollars. Since we have "x" half dollars then the value of these half dollars is x*0.50 = 0.5x
Similarly we can determine that...
the value of 2x quarters is (2x)*0.25 = 0.5x
the value of 4x dimes is (4x)*0.10 = 0.4x
the value of 8x nickels is (12x)*0.05 = 0.6x
We're now ready to write our equation and solve it. Again, the equation will express the fact the the sum of the values of all these coins is $10.00:
0.5x + 0.5x + 0.4x + 0.6x = 10
Simplifying by adding like terms on the left:
2x = 10
Divide both sides by 2:
x = 5
Now we can use our expressions to answer the question.
Since x represents the number of half dollars, she got 5 half dollars.
Since 2x represents the number of quarters, she got 2(5) or 10 quarters.
Since 4x represents the number of dimes, she got 4(5) or 20 dimes.
Since 12x represents the number of nickels, she got 12(5) or 60 nickels.