Fractions of an amount

Question

I need help with the following problem:

Catalin works in an office. One week he divides his time between these tasks:

• $\frac{1}{4}$ of his time in meetings
• $\frac{5}{8}$ of his time writing reports
• the rest of his time doing the accounts

He spends a total of $6$ hours doing the accounts

Find the total number of hours he works in the week. How would I go about solving this problem?

Answer 1

• $ \frac{1}{4}$ of time in meetings

• $ \frac{5}{8}$ of time writing reports

• rest of the time spending doing accounts

How to solve the problem

Firstly, you know how the individual separates his time between three tasks. You're given the fraction of the time he spends on two of the tasks. The sum of the fraction of times he spends on tasks should be equal to 1, which is represents all time spent working. Therefore you can calculate the fraction of time the individual spends doing accounts.

Now you have the fraction of the total time the individual spends doing accounts, which we know takes 6 hours. Use ratio's to calculate the total time spent working.

The solution

$1 - \frac{1}{4} - \frac{5}{8} = \frac{1}{8} $

$\frac{1}{8} $ of the total time is spent doing accounts, which we know takes 6 hours.

Therefore the total time spent working is $ 8*6 = 48 $

Answer 2

There's another approach, which may have value, depending on your priorities with regard to understanding one thing quickly as opposed to developing your math chops. (I'm not ranking one above the other, mind you, but they are different objectives.)

That approach is guessing. One can get better at guessing, and in doing so, develop an intuition as to which way solutions will "go". In this particular instance, we might start out by guessing a $40$-hour work week. We then try that guess out. One-fourth of $40$ is $10$ hours in meetings (gak), and five-eighths of $40$ is $25$ hours writing reports (double gak). That leaves $40-10-25 = 5$ hours left for doing accounts.

Obviously, that didn't work, but it was close, so we might anticipate that if there is a solution, it will be close to $40$ hours. So let's try another number; let's try $35$. One-fourth of $35$ is $8.75$ hours spent in meetings; five-eighths of $35$ is $21.875$ hours spent writing reports. That leaves $35-8.75-21.875 = 4.375$ hours left for doing accounts.

Two lessons from this second trial: one, we reduced the total hours, and we had fewer hours left for doing accounts, where we wanted more; and two, it's easier if we guess numbers divisible by $4$ and/or $8$. So our next guess will be more than $40$, and it will be a "nice" number, where $4$ and $8$ are concerned. We can guess $44$ or $48$, and one of those will get us what we want.

To be sure, this approach does not give the "right" analysis, and if we don't provide that, we (meaning you) probably won't get full credit on an actual problem. But if one pays close attention to how the various parts of the problem vary with each other, one can suss out how to analyze it, and better internalize the situation to boot.

It's trivial, once you have done a few of these problems, to see that you let the total numbers be denoted by $x$, that you express the variable parts in terms of $x$, and the constant parts as numbers, and you set up an equation. But without a recipe book in the form of a textbook or other aid (a situation that is common in the real world), we have to figure out how to approach problems, and one effective way to develop that skill, in my opinion, is to practice guessing.