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# Tricky Word Problem

This word problem has me stumped..  I would really lovesome step by step help, thanks!

When a number, half of a number, and a third of a number are added together, the sum is 385. Find the three numbers.

### 5 Answers by Expert Tutors

Jessica S. | Highly Effective Teacher seeks to create life-long learnersHighly Effective Teacher seeks to create...
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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 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
Benjamin M. | UC Student Specialized in Mathematics and Earth SciencesUC Student Specialized in Mathematics an...
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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.
Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
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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 √
Steve P. | Patient and Knowledgable Math and Chemistry TutorPatient and Knowledgable Math and Chemis...
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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

Vivian L. | Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACHMicrosoft Word/Excel/Outlook, essay comp...
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