Fraction Work: Solve & Reduce Hours Worked To Lowest Terms

Hey guys! Ever find yourself staring at a word problem involving fractions and feeling a bit overwhelmed? Don't worry, you're not alone! Fractions can seem tricky, but with a little practice, you'll be solving them like a pro. In this article, we'll break down a common type of problem – calculating work hours and expressing the answer in its simplest fractional form. We'll use a real-world example to guide you through each step, making the process clear and easy to follow.

Understanding the Problem: Marie's Two Jobs

Let's imagine this scenario: Marie worked a total of 40 hours across two different jobs. On her first job, she clocked in $4 \frac{1}{2}$ hours on Monday, $3 \frac{7}{8}$ hours on Tuesday, and 2 hours on both Wednesday and Thursday. Our mission, should we choose to accept it, is to figure out how many hours Marie worked on her second job and then express that time as a fraction in its simplest form. This means we need to not only find the answer but also make sure the fraction is reduced to its lowest terms. No sweat, we've got this!

Step 1: Calculate Total Hours on the First Job

First things first, let's calculate the total hours Marie dedicated to her first job. We know her hours for Monday, Tuesday, Wednesday, and Thursday, so we need to add them all up. This is where those fractions come into play, but don't let them scare you! We'll tackle them one step at a time.

Marie's hours on her first job break down like this:

• Monday: $4 \frac{1}{2}$ hours
• Tuesday: $3 \frac{7}{8}$ hours
• Wednesday: 2 hours
• Thursday: 2 hours

To get the total, we need to perform the addition: $4 \frac{1}{2} + 3 \frac{7}{8} + 2 + 2$. The key to adding mixed numbers like these is to first convert them into improper fractions. This makes the addition process much smoother. Remember, an improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

Let's convert $4 \frac{1}{2}$ into an improper fraction. We multiply the whole number (4) by the denominator (2) and add the numerator (1), then place the result over the original denominator. So, (4 * 2) + 1 = 9, giving us $\frac{9}{2}$.

Now, let's do the same for $3 \frac{7}{8}$. We multiply 3 by 8 and add 7, which gives us (3 * 8) + 7 = 31. This makes our improper fraction $\frac{31}{8}$.

Our equation now looks like this: $\frac{9}{2} + \frac{31}{8} + 2 + 2$. We can easily add the whole numbers (2 + 2 = 4), but to add the fractions, they need a common denominator. This is the magic number that allows us to combine the fractions seamlessly.

Step 2: Finding a Common Denominator

The key to adding fractions lies in finding a common denominator. This means finding a number that both denominators (in our case, 2 and 8) can divide into evenly. Think of it like finding a common language for the fractions to speak! The easiest way to find a common denominator is to look for the least common multiple (LCM) of the denominators. In simpler terms, it's the smallest number that both denominators can divide into without leaving a remainder.

In our problem, we have denominators of 2 and 8. What's the smallest number that both 2 and 8 can divide into? Well, 8 itself works perfectly! 8 divided by 2 is 4, and 8 divided by 8 is 1. So, 8 is our common denominator. This makes our lives much easier, as we only need to convert one fraction.

We already have $\frac{31}{8}$, which is great! Now we need to convert $\frac{9}{2}$ into an equivalent fraction with a denominator of 8. To do this, we ask ourselves: what do we multiply 2 by to get 8? The answer is 4. So, we multiply both the numerator (9) and the denominator (2) of $\frac{9}{2}$ by 4. This gives us $\frac{9 * 4}{2 * 4} = \frac{36}{8}$.

Now our equation looks even better: $\frac{36}{8} + \frac{31}{8} + 4$. We're ready to add those fractions!

Step 3: Adding Fractions and Whole Numbers

Now that we have a common denominator, adding the fractions is a breeze. We simply add the numerators and keep the denominator the same. So, $\frac{36}{8} + \frac{31}{8} = \frac{36 + 31}{8} = \frac{67}{8}$.

Don't forget about the whole number 4 we had earlier! We need to add this to our fraction sum. So, we have $\frac{67}{8} + 4$. To add a whole number to a fraction, it's helpful to think of the whole number as a fraction with a denominator of 1. So, 4 is the same as $\frac{4}{1}$.

But to add it to $\frac{67}{8}$, we need a common denominator again! We'll use 8 as our common denominator. To convert $\frac{4}{1}$ to a fraction with a denominator of 8, we multiply both the numerator and denominator by 8: $\frac{4 * 8}{1 * 8} = \frac{32}{8}$.

Now we can add: $\frac{67}{8} + \frac{32}{8} = \frac{67 + 32}{8} = \frac{99}{8}$.

So, Marie worked a total of $\frac{99}{8}$ hours on her first job. But this is an improper fraction, which can be a bit hard to visualize. Let's convert it back to a mixed number.

Step 4: Converting Improper Fractions to Mixed Numbers

To convert an improper fraction back to a mixed number, we divide the numerator by the denominator. The quotient (the result of the division) becomes the whole number part of our mixed number, the remainder becomes the numerator of the fractional part, and the denominator stays the same.

Let's divide 99 by 8. 8 goes into 99 eleven times (11 * 8 = 88), with a remainder of 11 (99 - 88 = 11). So, our mixed number is 12 \frac{3}{8}.

Therefore, Marie worked a total of $12 \frac{3}{8}$ hours on her first job. We're halfway there! Now we need to figure out how many hours she worked on her second job.

Step 5: Calculate Hours on the Second Job

We know Marie worked a total of 40 hours across both jobs. We also know she worked $12 \frac{3}{8}$ hours on her first job. To find the hours she worked on her second job, we simply subtract the hours from the first job from the total hours.

So, we need to calculate $40 - 12 \frac{3}{8}$. This might look a bit tricky, but we can handle it! First, let's convert the mixed number $12 \frac{3}{8}$ back into an improper fraction. We already know how to do this: (12 * 8) + 3 = 99, so $12 \frac{3}{8} = \frac{99}{8}$.

Now our equation is $40 - \frac{99}{8}$. To subtract a fraction from a whole number, we need to express the whole number as a fraction with the same denominator as the fraction we're subtracting. So, we need to turn 40 into a fraction with a denominator of 8. We do this by multiplying 40 by $\frac{8}{8}$ (which is just 1, so we're not changing the value): $40 * \frac{8}{8} = \frac{40 * 8}{8} = \frac{320}{8}$.

Now we can subtract: $\frac{320}{8} - \frac{99}{8} = \frac{320 - 99}{8} = \frac{221}{8}$.

Marie worked $\frac{221}{8}$ hours on her second job. But we're not quite done yet! We need to express this as a mixed number and then reduce it to its lowest terms.

Step 6: Reduce the Fraction to Lowest Terms

We've already got our answer as an improper fraction: $\frac{221}{8}$. Let's first convert this to a mixed number. We divide 221 by 8. 8 goes into 221 twenty-seven times (27 * 8 = 216) with a remainder of 5 (221 - 216 = 5). So, our mixed number is $27 \frac{5}{8}$.

Now comes the final step: reducing the fraction to its lowest terms. This means finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it. The GCF is the largest number that divides evenly into both the numerator and the denominator.

In our fraction, $\frac{5}{8}$, the numerator is 5 and the denominator is 8. The factors of 5 are 1 and 5. The factors of 8 are 1, 2, 4, and 8. The only common factor they share is 1. This means that $\frac{5}{8}$ is already in its simplest form! We can't reduce it any further.

The Final Answer

So, after all our calculations, we've arrived at the solution! Marie worked $27 \frac{5}{8}$ hours on her second job. The fraction $\frac{5}{8}$ is already in its lowest terms, so we're done! Woohoo!

Key Takeaways for Fraction Mastery

• Converting Mixed Numbers to Improper Fractions: This simplifies addition and subtraction. Multiply the whole number by the denominator and add the numerator, then put the result over the original denominator.
• Finding a Common Denominator: This is crucial for adding and subtracting fractions. The least common multiple (LCM) of the denominators is your best bet.
• Adding and Subtracting Fractions: Once you have a common denominator, simply add or subtract the numerators and keep the denominator the same.
• Converting Improper Fractions to Mixed Numbers: Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same.
• Reducing Fractions to Lowest Terms: Find the greatest common factor (GCF) of the numerator and denominator and divide both by it.

Practice Makes Perfect

Fractions might seem daunting at first, but with consistent practice, you'll become a fraction-solving whiz! Try working through similar problems and focusing on understanding each step. Remember, the key is to break down the problem into smaller, manageable parts. You've got this!