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Tricky Word Problem

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When a number, half of a number, and a third of a number are added together, the sum is 385. Find the three numbers.
 
 
Let's make the mystery number represented as, "n," for 'number.'
 
We start with n.
 
 
We know we're going to be adding things together, so we can add a + sign after our "n."
 
n+
 
 
'Half of a number,' can be expressed by dividing 'n' in half. We can write this out as  n/2.
 
(Confused? Plug in a number! Four, for example, divided by two equals two. Half of four = 2)
 
 
 
'Third of a number," can be expressed by dividing 'n' into thirds. We can write this out as n/3. 
 
 
 
When we combine all three of these terms, we come up with:
 
 
 
n + (n/2) + (n/3)
 
The whole thing equals 385.
 
 
 
n + (n/2) + (n/3) = 385
 
 
 
 
As suggested in another answer, we can factor out n to deal with our fractions/decimals first. Factoring is simply pulling out pieces (multiples) of a larger number. Don't forget that there's an invisible "1" in front of variables (letters representing a mystery number).
 
 
n (1/1 + 1/2 + 1/3) = 385
 
Now we need to combine terms within the parenthesis. However, we can't add fractions until we've found a common denominator. Six is an easy number to use, since we can multiply (1/2) x 3/3 to get 3/6, and multiply (1/3) x 2/2 to get 2/6. And of course, 1/1 x 6/6 = 6/6.
 
 
n (6 + 3 + 2) = 385
    6    6    6
 
n (11) = 385
     6
 
 
Let's get rid of that ugly fraction. Multiply BOTH sides by 6.
 
(6) 11n = 385 (6)
       6
 
 
11n = 2310
 
 
 
To isolate n, we need to divide each side by 11.
 
11n = 2310
 11        11
 
 
We end up with:
 
n = 210
 
 
Great, now we know what n is, but what about n/2 and n/3? This is the easy part- just plug it in!
 
n = 210 = 105
2      2
 
n = 210 = 70
3     3
 
 
 
Let's check:
 
n + (n/2) + (n/3) = 385
 
210 + 105 + 70 = 385
 
385 = 385
This is a question about variables. It's asking you to substitute a variable for the "number" referred to in the problem.
 
Basically, you have to know how to put that "number" in an equation.
 
"a number" means x
"half of a number" means x times one half = x/2
"a third of a number" means x times one third = x/3
 
Once you know that, the equation sorts itself out
a number, half of a number, and a third of a number are added together, the sum is 385
means
x + (x/2) + (x/3) = 385
 
After lots of un-fun algebra you can solve for x
 
x + x/2 + x/3 = 385
x(1+1/2+1/3) = 385 <----factoring out x
 
now, 1 + 1/2 + 1/3 looks ugly, but its easier if we look at it this way
 
1/1 + 1/2 + 1/3
 
now it's a fraction addition problem
 
the common denominator of 1, 2, and 3 is 6, so
 
1/1 + 1/2 + 1/3 =
6/6 + 3/6 + 2/6 = 11/6
 
so 1 + 1/2 + 1/3 = 11/6 (let's leave it as an improper fraction for now.)
 
back to the problem:
 
x(11/6) = 385 <---to "cancel" 11/6, we multiply by 6, then divide by 11
6*x*(11/6) = 385*6
11x = 2310
11x/11 = 2310/11
x = 210
 
Great! But the question didnt' ask for x, it asked for the three numbers.
 
So, back to our original equation.
 
x + (x/2) + (x/3) = 385
 
If x = 210, then
x/2 means 210/2, or 105
and
x/3 means 210/3, or 70
So the "number" is 210, the "half of a number" is 105, and the "a third of a number" is 70.
When a number (x), half of a number (x/2), and a third of a number (x/3) are added together (x+x/2+x/3), the sum is 385 (x+x/2+x/3=385).
 
Let's multiply both sides of the equation by 2*3=6:
6x+2*3*(x/2)+2*3*(x/3)=6*385
 
Simplify; expressing 385 as a product of prime factors:
6x+3x+2x=6*5*7*11
11x=210*11
x=210
x/2 = 105
x/3 = 70
 
check: 210+105+70=385 √
x+1/2x+1/3x=385
FInd the common denominator so you can add the fractions:
 
6/6x+3/6x+2/6x=385
 
Multiply both sides by 6/11
 
6/11(11/6x)= 6/11(385)
 
x=210
 
1/2(210)=105
 
1/3(210)=70
 
The 3 numbers are 70,105 and 210
 
 
70+105+210=385
 
Hi Addie;
x+(x/2)+(x/3)=385
Let's multiply both sides by 6 (2 x 3) to eliminate the fractions...
6[x+(x/2)+(x/3)]=(385)(6)
6x+3x+2x=2310
11x=2310
Let's divide both sides by 11...
(11x)/11=2310/11
x=210
x/2=105
x/3=70
Let's check our work...
210+105+70=385
385=385