Math Problems: Fraction, Equation Solutions & Calculations

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Math Problems: Fraction, Equation Solutions & Calculations

Hey guys! Let's break down these math problems step-by-step. We'll tackle everything from fractions and basic arithmetic to solving equations. Get ready to sharpen those pencils (or keyboards!) and dive in. I know that math can be tedious sometimes, but I'll explain things in a way that is easy to understand.

Fraction and Mixed Number Arithmetic

Problem 1: Adding Mixed Numbers

Let's start with adding mixed numbers: 7 1/43 + 2 1/3. This problem involves adding two mixed numbers, and to do this effectively, we'll convert them into improper fractions first. Converting mixed numbers to improper fractions makes the addition process smoother, especially when dealing with unlike denominators.

So, 7 1/43 can be converted to an improper fraction as follows:

  • Multiply the whole number part (7) by the denominator (43): 7 * 43 = 301
  • Add the numerator (1) to the result: 301 + 1 = 302
  • Place this sum over the original denominator (43): 302/43

Similarly, 2 1/3 can be converted to an improper fraction:

  • Multiply the whole number part (2) by the denominator (3): 2 * 3 = 6
  • Add the numerator (1) to the result: 6 + 1 = 7
  • Place this sum over the original denominator (3): 7/3

Now, we have the expression 302/43 + 7/3. To add these fractions, we need a common denominator. The least common multiple (LCM) of 43 and 3 is 129 (since 43 is a prime number). Therefore, we convert each fraction to have this denominator:

  • For 302/43, multiply both the numerator and the denominator by 3: (302 * 3) / (43 * 3) = 906/129
  • For 7/3, multiply both the numerator and the denominator by 43: (7 * 43) / (3 * 43) = 301/129

Now, add the two fractions: 906/129 + 301/129 = 1207/129.

Finally, we convert the improper fraction back to a mixed number. To do this, divide 1207 by 129:

  • 1207 ÷ 129 = 9 with a remainder of 46. This means 1207/129 = 9 46/129.

So, 7 1/43 + 2 1/3 = 9 46/129. This comprehensive breakdown ensures clarity and accuracy in solving the problem.

Problem 2: Simplify the Expression

Next, let's tackle: 9 24/9 + 27. This looks like it could be a mixed number situation. However, 24/9 can be simplified and converted to a mixed number or improper fraction. Simplify the fraction and then handle the addition.

  • First, we simplify 24/9. Both 24 and 9 are divisible by 3. So, 24 ÷ 3 = 8 and 9 ÷ 3 = 3. Thus, 24/9 simplifies to 8/3.
  • Now, 8/3 can be written as a mixed number: 2 2/3 because 8 divided by 3 is 2 with a remainder of 2.
  • So, we can rewrite the original expression as 9 + 2 2/3 + 27.
  • Next, let's combine the whole numbers: 9 + 2 + 27 = 38.
  • Therefore, the expression becomes 38 2/3.

Therefore, 9 24/9 + 27 = 38 2/3. This makes sure that the answer is both correct and in its simplest form.

Problem 3: Subtracting Fractions from Whole Numbers

Now, let's solve: 1 - 12/19. This problem involves subtracting a fraction from a whole number. To perform this subtraction, we need to express the whole number as a fraction with the same denominator as the fraction being subtracted. By converting the whole number into a fraction with a common denominator, we can easily subtract the fractions.

Here’s how we do it:

  • We start with 1 - 12/19.
  • To subtract 12/19 from 1, we need to express 1 as a fraction with the denominator 19. So, we rewrite 1 as 19/19.
  • Now, the expression becomes 19/19 - 12/19.
  • Subtract the numerators: 19 - 12 = 7.
  • Place the result over the common denominator: 7/19.

Therefore, 1 - 12/19 = 7/19. Understanding this process helps to efficiently subtract fractions from whole numbers. The result is now in its simplest form.

Problem 4: Subtracting Whole Numbers

Let's evaluate 8 - 36. This problem involves subtracting a larger number from a smaller number, which will result in a negative number. Be mindful of the negative sign! This problem is pretty straight forward.

  • Simply perform the subtraction: 8 - 36 = -28

So, 8 - 36 = -28. Make sure to pay attention to the order of subtraction, especially when dealing with negative numbers.

Problem 5: Subtracting Mixed Numbers from Whole Numbers

Now let's solve: 12 - 11 1/3. When you're subtracting a mixed number from a whole number, it's often easiest to turn the mixed number into an improper fraction first. This makes the subtraction process more straightforward and less prone to errors. Convert the mixed number to an improper fraction, find a common denominator, and then subtract.

  • First, convert the mixed number 11 1/3 to an improper fraction.
  • Multiply the whole number (11) by the denominator (3): 11 * 3 = 33
  • Add the numerator (1) to the result: 33 + 1 = 34
  • Place this sum over the original denominator (3): 34/3
  • Now, the expression becomes 12 - 34/3.
  • To subtract 34/3 from 12, we need to express 12 as a fraction with the denominator 3. So, we rewrite 12 as 36/3 because 12 * 3 = 36.
  • Now, the expression is 36/3 - 34/3.
  • Subtract the numerators: 36 - 34 = 2.
  • Place the result over the common denominator: 2/3.

Therefore, 12 - 11 1/3 = 2/3. This makes the computation more manageable and accurate.

Problem 6: Multiplying Fractions and Whole Numbers

Evaluate: 16/8 * 6/13 * 13/15 * 1219097. This involves multiplication of fractions and a whole number. To solve this efficiently, simplify the fractions first and then multiply. This simplifies the calculations and helps to easily reach the final answer.

  • We have: 16/8 * 6/13 * 13/15 * 1219097
  • Simplify 16/8 to 2. So, the expression becomes 2 * 6/13 * 13/15 * 1219097.
  • Notice that we are multiplying by 6/13 and 13/15. The 13 in the denominator of 6/13 cancels with the 13 in the numerator of 13/15. So, we have 2 * 6/15 * 1219097.
  • Simplify 6/15 by dividing both numerator and denominator by 3, giving 2/5.
  • Now, we have 2 * 2/5 * 1219097 = 4/5 * 1219097
  • Multiply 4/5 by 1219097: (4 * 1219097) / 5 = 4876388 / 5.
  • Now, divide 4876388 by 5 to get 975277.6.

Therefore, 16/8 * 6/13 * 13/15 * 1219097 = 975277.6. By simplifying fractions and systematically multiplying, we arrive at the final result.

Problem 7: Mixed Numbers and Improper Fractions

Calculate: 13 14/9 - 29/3. This problem involves subtracting an improper fraction from a mixed number. As with addition, converting mixed numbers to improper fractions can make the subtraction process clearer and more accurate.

  • Convert the mixed number 13 14/9 to an improper fraction.
  • Multiply the whole number part (13) by the denominator (9): 13 * 9 = 117
  • Add the numerator (14) to the result: 117 + 14 = 131
  • Place the sum over the original denominator (9): 131/9
  • Now, the expression is 131/9 - 29/3.
  • To subtract 29/3 from 131/9, we need a common denominator. The least common multiple (LCM) of 9 and 3 is 9. So, we need to convert 29/3 to have a denominator of 9. Multiply both the numerator and the denominator of 29/3 by 3: (29 * 3) / (3 * 3) = 87/9.
  • Now, the expression is 131/9 - 87/9.
  • Subtract the numerators: 131 - 87 = 44.
  • Place the result over the common denominator: 44/9.
  • Convert the improper fraction 44/9 back to a mixed number. Divide 44 by 9: 44 ÷ 9 = 4 with a remainder of 8. So, 44/9 = 4 8/9.

Thus, 13 14/9 - 29/3 = 4 8/9. Breaking the problem down into smaller steps ensures an accurate and understandable solution.

Problem 8: Subtraction with Mixed Numbers and Whole Numbers

Let's evaluate: 107 - 4 12/16. This problem involves subtracting a mixed number from a whole number. Simplify the mixed number, convert it to an improper fraction, find a common denominator, and subtract.

  • We start with 107 - 4 12/16.
  • First, simplify the fraction 12/16 by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So, 12 ÷ 4 = 3 and 16 ÷ 4 = 4. The fraction simplifies to 3/4.
  • Now, the mixed number is 4 3/4.
  • Convert the mixed number 4 3/4 to an improper fraction. Multiply the whole number part (4) by the denominator (4): 4 * 4 = 16. Add the numerator (3) to the result: 16 + 3 = 19. Place the sum over the original denominator (4): 19/4.
  • The expression is now 107 - 19/4.
  • To subtract 19/4 from 107, we need to express 107 as a fraction with the denominator 4. So, we rewrite 107 as 428/4 because 107 * 4 = 428.
  • Now, the expression is 428/4 - 19/4.
  • Subtract the numerators: 428 - 19 = 409.
  • Place the result over the common denominator: 409/4.
  • Convert the improper fraction 409/4 back to a mixed number. Divide 409 by 4: 409 ÷ 4 = 102 with a remainder of 1. So, 409/4 = 102 1/4.

Therefore, 107 - 4 12/16 = 102 1/4. Simplifying each step ensures accuracy and ease of comprehension.

Problem 9: Subtraction with Mixed Numbers

Calculate 29 49/53 - 8 49/53. Here, we're subtracting two mixed numbers that have the same fractional part (49/53). This makes things a bit easier, as we can focus on subtracting the whole number parts first. Simplify by subtracting whole numbers first and then deal with any fractional parts.

  • We start with 29 49/53 - 8 49/53.
  • Subtract the whole numbers: 29 - 8 = 21.
  • Since the fractional parts are the same (49/53), they cancel out: 49/53 - 49/53 = 0.
  • The result is just the difference of the whole numbers.

Therefore, 29 49/53 - 8 49/53 = 21. This simplified approach highlights the efficiency of subtracting mixed numbers with identical fractional parts.

Problem 10: Mixed Numbers and Order of Operations

Evaluate: (2016 + 13/14). This problem combines a whole number and a fraction inside parentheses. The key here is to follow the order of operations, which dictates that we perform the operation inside the parentheses first.

  • Inside the parentheses, we have 2016 + 13/14. To add these, we need to express 2016 as a fraction with a denominator of 14. So, 2016 = 2016/1.
  • Multiply the numerator and denominator of 2016/1 by 14 to get a common denominator: (2016 * 14) / (1 * 14) = 28224/14.
  • Now, the expression inside the parentheses is 28224/14 + 13/14.
  • Add the numerators: 28224 + 13 = 28237.
  • Place the result over the common denominator: 28237/14.
  • Since the problem asks us to evaluate and doesn't specify a mixed number, we can leave it as an improper fraction.

Therefore, (2016 + 13/14) = 28237/14. Understanding and applying the order of operations ensures the correct solution. Alternatively, you could convert it into a mixed number if needed: 2016 13/14.

Solving Equations

Equation 1: Solving for x

Solve for x: x + 4/19 = 62/19. This is a basic algebraic equation. To find the value of x, we need to isolate x on one side of the equation by subtracting 4/19 from both sides. Isolating the variable is the fundamental step in solving any algebraic equation.

  • Starting with: x + 4/19 = 62/19
  • Subtract 4/19 from both sides of the equation: x + 4/19 - 4/19 = 62/19 - 4/19
  • This simplifies to: x = 62/19 - 4/19
  • Since the denominators are the same, we can directly subtract the numerators: x = (62 - 4) / 19
  • So, x = 58/19

Therefore, x = 58/19. If you want to express it as a mixed number, x = 3 1/19. This makes the process clear and accurate.

Equation 2: Solving for x with Subtraction

Solve for x: 25 - x = 8/3. To isolate x in this equation, we'll need to rearrange the equation. The most straightforward approach is to subtract 25 from both sides and then multiply by -1 to solve for x. Rearranging the equation correctly is vital to find the correct value of x.

  • Starting with: 25 - x = 8/3
  • Subtract 25 from both sides: -x = 8/3 - 25
  • Convert 25 to a fraction with a denominator of 3: 25 = 75/3
  • Now, the equation is: -x = 8/3 - 75/3
  • Subtract the fractions: -x = (8 - 75) / 3
  • Simplify: -x = -67/3
  • Multiply both sides by -1 to solve for x: x = 67/3

Therefore, x = 67/3. If you want to express it as a mixed number, x = 22 1/3. This step-by-step approach ensures clarity and correctness.

Equation 3: Isolate the variable x

Solve for x: 5/9 = 13. This looks like a strange equation since x is not in the equation. There is no variable x, so there is no way to solve for x. Therefore, there is no solution.

So, 5/9 = 13 has no solution because x is not in the equation.

I hope that these steps helped you to understand how to solve each problem. Let me know if you have any other math problems.