Solving Numerical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into some cool math problems. We're going to tackle numerical expressions, breaking them down step by step to make sure we understand everything. This is a great way to brush up on your arithmetic skills and get a solid grasp of how these expressions work. So, grab your pencils and let's get started! We will be going over several numerical expressions and calculating their values, using the correct order of operations, and simplifying fractions. This guide will take you through each problem, explaining the process clearly. I'll include all the steps so you don't miss anything. If you're feeling a bit rusty or just want to sharpen your math skills, then you've come to the right place. Don't worry, it's not as hard as it looks! We'll start with the basics, like fractions, mixed numbers, and order of operations. Then we'll move on to some more complex expressions. Remember, the key to solving these types of problems is to take it one step at a time. Let's make math fun and easy together!

Expression 1: Decoding the First Numerical Puzzle

Let's start with our first expression: $\frac{3}{5}:\frac{9}{10}+3\frac{3}{4}:2\frac{2}{5}-4:2\frac{2}{3}$. This one looks a bit intimidating at first glance, but let's break it down into smaller, more manageable pieces. The key here is to follow the order of operations, which you might remember as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this case, we'll be dealing with division, addition, and subtraction. First, we need to convert the mixed numbers into improper fractions to make the division easier. Remember that a mixed number like $3\frac{3}{4}$ is the same as $(3 \times 4) + 3$, all over 4, which equals $\frac{15}{4}$. Similarly, $2\frac{2}{5}$ becomes $\frac{12}{5}$, and $2\frac{2}{3}$ is $\frac{8}{3}$. Now we can rewrite the expression as $\frac{3}{5}:\frac{9}{10}+\frac{15}{4}:\frac{12}{5}-4:\frac{8}{3}$. Next, let's tackle the division operations. Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, $\frac{3}{5}:\frac{9}{10}$ becomes $\frac{3}{5} \times \frac{10}{9}$. Then, $\frac{15}{4}:\frac{12}{5}$ becomes $\frac{15}{4} \times \frac{5}{12}$, and $4:\frac{8}{3}$ is the same as $\frac{4}{1}:\frac{8}{3}$, which turns into $\frac{4}{1} \times \frac{3}{8}$. Now we have $\frac{3}{5} \times \frac{10}{9} + \frac{15}{4} \times \frac{5}{12} - \frac{4}{1} \times \frac{3}{8}$. Let's simplify these multiplications. $\frac{3}{5} \times \frac{10}{9}$ simplifies to $\frac{2}{3}$, $\frac{15}{4} \times \frac{5}{12}$ simplifies to $\frac{25}{16}$, and $\frac{4}{1} \times \frac{3}{8}$ simplifies to $\frac{3}{2}$. Finally, our expression is now $\frac{2}{3} + \frac{25}{16} - \frac{3}{2}$. To add and subtract these fractions, we need a common denominator. The least common denominator (LCD) of 3, 16, and 2 is 48. Convert each fraction to have a denominator of 48: $\frac{2}{3}$ becomes $\frac{32}{48}$, $\frac{25}{16}$ becomes $\frac{75}{48}$, and $\frac{3}{2}$ becomes $\frac{72}{48}$. Our expression is now $\frac{32}{48} + \frac{75}{48} - \frac{72}{48}$. Adding and subtracting, we get $\frac{32+75-72}{48}$, which simplifies to $\frac{35}{48}$. So, the value of the first expression is $\frac{35}{48}$.

Step-by-Step Breakdown

1. Convert mixed numbers to improper fractions:
   • $3\frac{3}{4} = \frac{15}{4}$
   • $2\frac{2}{5} = \frac{12}{5}$
   • $2\frac{2}{3} = \frac{8}{3}$
2. Rewrite the expression:
   • $\frac{3}{5}:\frac{9}{10} + \frac{15}{4}:\frac{12}{5} - 4:\frac{8}{3}$
3. Perform division by multiplying by the reciprocal:
   • $\frac{3}{5} \times \frac{10}{9} + \frac{15}{4} \times \frac{5}{12} - \frac{4}{1} \times \frac{3}{8}$
4. Simplify the multiplications:
   • $\frac{2}{3} + \frac{25}{16} - \frac{3}{2}$
5. Find the least common denominator (LCD): LCD is 48.
6. Convert fractions to have the LCD:
   • $\frac{32}{48} + \frac{75}{48} - \frac{72}{48}$
7. Add and subtract the fractions:
   • $\frac{35}{48}$

Expression 2: Unraveling the Second Numerical Expression

Alright, let's move on to the second expression: $7\frac{1}{3}:12\frac{1}{4}\cdot6\frac{1}{8}$. This one involves a mix of division and multiplication. Just like before, we'll start by converting the mixed numbers into improper fractions. Remember, the mixed number $7\frac{1}{3}$ becomes $\frac{22}{3}$, $12\frac{1}{4}$ becomes $\frac{49}{4}$, and $6\frac{1}{8}$ becomes $\frac{49}{8}$. Now our expression looks like this: $\frac{22}{3}:\frac{49}{4} \cdot \frac{49}{8}$. Let's handle the division first. We know that dividing by a fraction is the same as multiplying by its reciprocal. So, $\frac{22}{3}:\frac{49}{4}$ becomes $\frac{22}{3} \times \frac{4}{49}$. Now our expression is $\frac{22}{3} \times \frac{4}{49} \cdot \frac{49}{8}$. We can then perform the multiplication operations. Multiply the fractions: $\frac{22}{3} \times \frac{4}{49} \times \frac{49}{8}$. Before multiplying everything out, let's see if we can simplify. Notice that there's a 49 in both the numerator and denominator, which cancels each other out. And, there is a common factor of 2 between 22 and 8, then we get $\frac{11}{3} \times \frac{1}{1} \times \frac{1}{1}$. So now we have $\frac{11}{3} \times \frac{1}{2}$. Finally, multiplying the remaining fractions, we get $\frac{11}{3} \times \frac{1}{2}$, which is $\frac{11}{6}$. We can convert this improper fraction back into a mixed number, which is $1\frac{5}{6}$. Therefore, the value of the second expression is $1\frac{5}{6}$. See? Not so bad, right?

Step-by-Step Breakdown

1. Convert mixed numbers to improper fractions:
   • $7\frac{1}{3} = \frac{22}{3}$
   • $12\frac{1}{4} = \frac{49}{4}$
   • $6\frac{1}{8} = \frac{49}{8}$
2. Rewrite the expression:
   • $\frac{22}{3}:\frac{49}{4} \cdot \frac{49}{8}$
3. Perform division by multiplying by the reciprocal:
   • $\frac{22}{3} \times \frac{4}{49} \cdot \frac{49}{8}$
4. Simplify and multiply the fractions:
   • $\frac{22}{3} \times \frac{4}{49} \cdot \frac{49}{8} = \frac{11}{3} \times \frac{1}{1} \times \frac{1}{2}$
   • $\frac{11}{6} = 1\frac{5}{6}$

Expression 3: Solving the Third Expression

Let's wrap things up with our final expression: $1\frac{5}{12}:\frac{3}{34}+1\frac{1}{12}\cdot1\frac{31}{34}$. This one has a mix of division and multiplication, too. Again, the first thing we want to do is convert those mixed numbers into improper fractions. The mixed number $1\frac{5}{12}$ converts to $\frac{17}{12}$, and $1\frac{1}{12}$ becomes $\frac{13}{12}$, and $1\frac{31}{34}$ becomes $\frac{65}{34}$. So, our expression now looks like this: $\frac{17}{12}:\frac{3}{34}+\frac{13}{12}\cdot\frac{65}{34}$. Let's handle the division first. Dividing by a fraction means multiplying by its reciprocal, so $\frac{17}{12}:\frac{3}{34}$ becomes $\frac{17}{12} \times \frac{34}{3}$. Now we have $\frac{17}{12} \times \frac{34}{3} + \frac{13}{12} \cdot \frac{65}{34}$. Perform the multiplication from left to right. Now we have $\frac{17}{12} \times \frac{34}{3} = \frac{289}{18}$ and $\frac{13}{12} \cdot \frac{65}{34} = \frac{845}{408}$. So our expression becomes $\frac{289}{18} + \frac{845}{408}$. To add these fractions, we need a common denominator. The least common denominator (LCD) of 18 and 408 is 2448. Now, convert both fractions to have a denominator of 2448. $\frac{289}{18}$ becomes $\frac{39304}{2448}$, and $\frac{845}{408}$ becomes $\frac{5070}{2448}$. Then, we can add the fractions: $\frac{39304}{2448} + \frac{5070}{2448} = \frac{39304+5070}{2448}$. Adding the fractions gives us $\frac{44374}{2448}$. Simplify the fraction to $\frac{22187}{1224}$. We can also write this as a mixed number: $18\frac{235}{1224}$. Nice job, guys! You did it!

Step-by-Step Breakdown

1. Convert mixed numbers to improper fractions:
   • $1\frac{5}{12} = \frac{17}{12}$
   • $1\frac{1}{12} = \frac{13}{12}$
   • $1\frac{31}{34} = \frac{65}{34}$
2. Rewrite the expression:
   • $\frac{17}{12}:\frac{3}{34} + \frac{13}{12} \cdot \frac{65}{34}$
3. Perform division by multiplying by the reciprocal:
   • $\frac{17}{12} \times \frac{34}{3} + \frac{13}{12} \cdot \frac{65}{34}$
4. Multiply the fractions:
   • $\frac{289}{18} + \frac{845}{408}$
5. Find the least common denominator (LCD): LCD is 2448.
6. Convert fractions to have the LCD:
   • $\frac{39304}{2448} + \frac{5070}{2448}$
7. Add the fractions:
   • $\frac{44374}{2448} = \frac{22187}{1224} = 18\frac{235}{1224}$

Conclusion: You've Got This!

Great job working through these numerical expressions, everyone! Remember, the key is to take it one step at a time, follow the order of operations, and simplify your fractions. Keep practicing, and you'll become a pro in no time! Keep up the awesome work!