Fraction Word Problems: Solve Distance & Difference

Hey guys! Today, we're diving into some word problems involving fractions. Don't worry, we'll break it down so it's super easy to understand. We've got two main problems here: one about adding 12 to the difference of two mixed numbers, and another about Akmal's running distances. Let's get started!

Adding 12 to the Difference of Mixed Numbers

So, the first part of our challenge is: add 12 to the difference between $2\frac{5}{6}$ and $1\frac{1}{4}$. Sounds a bit complicated, but we'll take it one step at a time. The key here is to remember our order of operations and how to work with mixed numbers and fractions.

First things first, we need to find the difference between $2\frac{5}{6}$ and $1\frac{1}{4}$. This means we're subtracting the second mixed number from the first. To do this, it's helpful to convert these mixed numbers into improper fractions. This makes the subtraction process much smoother.

Let's convert $2\frac{5}{6}$ into an improper fraction. We multiply the whole number (2) by the denominator (6) and add the numerator (5). This gives us (2 * 6) + 5 = 17. We then put this result over the original denominator, so we have $\frac{17}{6}$.

Now, let's do the same for $1\frac{1}{4}$. Multiply the whole number (1) by the denominator (4) and add the numerator (1). This gives us (1 * 4) + 1 = 5. Place this over the original denominator to get $\frac{5}{4}$.

Okay, now we need to subtract $\frac{5}{4}$ from $\frac{17}{6}$. But, we can't subtract fractions directly unless they have a common denominator. So, we need to find the least common multiple (LCM) of 6 and 4. The LCM of 6 and 4 is 12. This means we need to convert both fractions to have a denominator of 12.

To convert $\frac{17}{6}$ to a fraction with a denominator of 12, we need to multiply both the numerator and the denominator by the same number. In this case, we multiply by 2 because 6 * 2 = 12. So, $\frac{17}{6}$ becomes $\frac{17 * 2}{6 * 2}$ which simplifies to $\frac{34}{12}$.

Similarly, to convert $\frac{5}{4}$ to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3 because 4 * 3 = 12. So, $\frac{5}{4}$ becomes $\frac{5 * 3}{4 * 3}$ which simplifies to $\frac{15}{12}$.

Now we can subtract! $\frac{34}{12} - \frac{15}{12} = \frac{34 - 15}{12} = \frac{19}{12}$.

So, the difference between $2\frac{5}{6}$ and $1\frac{1}{4}$ is $\frac{19}{12}$. This is an improper fraction, which we can convert back into a mixed number. To do this, we divide 19 by 12. 19 divided by 12 is 1 with a remainder of 7. So, $\frac{19}{12}$ is equal to $1\frac{7}{12}$.

We're not done yet! The original problem asked us to add 12 to this difference. So, we need to add 12 to $1\frac{7}{12}$. This is pretty straightforward. We just add the whole numbers together: 12 + 1 = 13. The fractional part stays the same. So, the final answer is $13\frac{7}{12}$.

Therefore, when you add 12 to the difference between $2\frac{5}{6}$ and $1\frac{1}{4}$, you get $13\frac{7}{12}$.

Akmal's Run: Distance Problems

Next up, we have a problem about Akmal's running distances. Akmal ran $2\frac{2}{5}$ km from point A to point B, and then $3\frac{7}{8}$ km from point B to point C. We have two questions to answer:

(a) What total distance did Akmal run?

This question is asking us to find the total distance, which means we need to add the two distances together. Akmal ran $2\frac{2}{5}$ km and $3\frac{7}{8}$ km, so we need to add these two mixed numbers.

Just like before, it's easiest to convert these mixed numbers into improper fractions first. Let's start with $2\frac{2}{5}$. Multiply the whole number (2) by the denominator (5) and add the numerator (2). This gives us (2 * 5) + 2 = 12. Put this over the original denominator to get $\frac{12}{5}$.

Now, let's convert $3\frac{7}{8}$. Multiply the whole number (3) by the denominator (8) and add the numerator (7). This gives us (3 * 8) + 7 = 31. Place this over the original denominator to get $\frac{31}{8}$.

So, we need to add $\frac{12}{5}$ and $\frac{31}{8}$. Again, we need a common denominator to add fractions. The least common multiple (LCM) of 5 and 8 is 40. So, we need to convert both fractions to have a denominator of 40.

To convert $\frac{12}{5}$ to a fraction with a denominator of 40, we multiply both the numerator and the denominator by 8 because 5 * 8 = 40. So, $\frac{12}{5}$ becomes $\frac{12 * 8}{5 * 8}$ which simplifies to $\frac{96}{40}$.

To convert $\frac{31}{8}$ to a fraction with a denominator of 40, we multiply both the numerator and the denominator by 5 because 8 * 5 = 40. So, $\frac{31}{8}$ becomes $\frac{31 * 5}{8 * 5}$ which simplifies to $\frac{155}{40}$.

Now we can add! $\frac{96}{40} + \frac{155}{40} = \frac{96 + 155}{40} = \frac{251}{40}$.

This is an improper fraction. Let's convert it back to a mixed number. 251 divided by 40 is 6 with a remainder of 11. So, $\frac{251}{40}$ is equal to $6\frac{11}{40}$.

Therefore, the total distance Akmal ran is $6\frac{11}{40}$ km.

(b) How much farther is BC than AB?

This question is asking us to find the difference between the distance from B to C and the distance from A to B. We already know Akmal ran $3\frac{7}{8}$ km from B to C and $2\frac{2}{5}$ km from A to B. So, we need to subtract the distance from A to B from the distance from B to C.

We've already converted these mixed numbers to improper fractions: $3\frac{7}{8}$ is $\frac{31}{8}$ and $2\frac{2}{5}$ is $\frac{12}{5}$. We also found a common denominator of 40 for these fractions.

So, we need to subtract $\frac{12}{5}$ (which is $\frac{96}{40}$) from $\frac{31}{8}$ (which is $\frac{155}{40}$). This is $\frac{155}{40} - \frac{96}{40} = \frac{155 - 96}{40} = \frac{59}{40}$.

Let's convert this improper fraction back to a mixed number. 59 divided by 40 is 1 with a remainder of 19. So, $\frac{59}{40}$ is equal to $1\frac{19}{40}$.

Therefore, the distance from B to C is $1\frac{19}{40}$ km farther than the distance from A to B.

Wrapping Up

Alright guys, we made it through those word problems! Remember, the key to solving fraction problems is to take them step by step. Convert mixed numbers to improper fractions, find common denominators, and don't be afraid to convert back to mixed numbers at the end. You got this!