4.2. Calculate: (Do Not Use A Calculator And Show All Working)${ 1 \frac{1}{2} + \frac{1}{4} \div \frac{3}{2} }$4.2.1. Thandu Is An Athlete. During A Practice Session, He Completes One Lap Of A Track In An Average Time Of $[ 3

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4.2. Calculate: (Do not use a calculator and show all working)

4.2.1. Thandu's Track Time

Thandu is an athlete who participates in track events. During a practice session, he completes one lap of a track in an average time of 3 minutes and 30 seconds. To understand his performance, we need to calculate his speed and time taken to complete the lap.

Understanding the Problem

The problem requires us to calculate the result of an expression involving fractions and mixed numbers. We will use the order of operations (PEMDAS) to evaluate the expression step by step.

Breaking Down the Expression

The given expression is:

112+14Γ·32{ 1 \frac{1}{2} + \frac{1}{4} \div \frac{3}{2} }

To simplify this expression, we need to follow the order of operations:

  1. Evaluate the mixed number 1121 \frac{1}{2}.
  2. Divide 14\frac{1}{4} by 32\frac{3}{2}.
  3. Add the results of steps 1 and 2.

Step 1: Evaluate the Mixed Number

A mixed number is a combination of a whole number and a fraction. In this case, the mixed number is 1121 \frac{1}{2}. To evaluate this, we need to convert it to an improper fraction.

112=(1Γ—2)+12=32{ 1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2} }

Step 2: Divide 14\frac{1}{4} by 32\frac{3}{2}

To divide fractions, we need to invert the second fraction and multiply.

14Γ·32=14Γ—23=1Γ—24Γ—3=212=16{ \frac{1}{4} \div \frac{3}{2} = \frac{1}{4} \times \frac{2}{3} = \frac{1 \times 2}{4 \times 3} = \frac{2}{12} = \frac{1}{6} }

Step 3: Add the Results

Now that we have evaluated the mixed number and the division, we can add the results.

32+16{ \frac{3}{2} + \frac{1}{6} }

To add fractions, we need to have a common denominator. The least common multiple (LCM) of 2 and 6 is 6.

32=3Γ—32Γ—3=96{ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} }

Now we can add the fractions:

96+16=106{ \frac{9}{6} + \frac{1}{6} = \frac{10}{6} }

Simplifying the Result

The result 106\frac{10}{6} can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2.

106=10Γ·26Γ·2=53{ \frac{10}{6} = \frac{10 \div 2}{6 \div 2} = \frac{5}{3} }

Therefore, the final result is 53\frac{5}{3}.

Conclusion

In this problem, we calculated the result of an expression involving fractions and mixed numbers. We followed the order of operations (PEMDAS) and used the rules for adding and dividing fractions to simplify the expression. The final result is 53\frac{5}{3}.

Key Takeaways

  • To evaluate a mixed number, convert it to an improper fraction.
  • To divide fractions, invert the second fraction and multiply.
  • To add fractions, have a common denominator and add the numerators.
  • To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

Practice Problems

  1. Evaluate the expression: 213+12Γ·342 \frac{1}{3} + \frac{1}{2} \div \frac{3}{4}
  2. Simplify the fraction: 1216\frac{12}{16}
  3. Add the fractions: 14+16+18\frac{1}{4} + \frac{1}{6} + \frac{1}{8}

Solutions

  1. 213+12Γ·34=73+23=93=32 \frac{1}{3} + \frac{1}{2} \div \frac{3}{4} = \frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3
  2. 1216=12Γ·416Γ·4=34\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}
  3. 14+16+18=624+424+324=1324\frac{1}{4} + \frac{1}{6} + \frac{1}{8} = \frac{6}{24} + \frac{4}{24} + \frac{3}{24} = \frac{13}{24}
    4.2. Calculate: (Do not use a calculator and show all working)

4.2.2. Q&A: Thandu's Track Time

In this section, we will answer some frequently asked questions related to Thandu's track time and the calculation of the expression.

Q: What is the average time taken by Thandu to complete one lap of the track?

A: The average time taken by Thandu to complete one lap of the track is 3 minutes and 30 seconds.

Q: How do we evaluate the mixed number 1121 \frac{1}{2}?

A: To evaluate the mixed number 1121 \frac{1}{2}, we need to convert it to an improper fraction. The mixed number 1121 \frac{1}{2} can be written as (1Γ—2)+12=32\frac{(1 \times 2) + 1}{2} = \frac{3}{2}.

Q: How do we divide fractions?

A: To divide fractions, we need to invert the second fraction and multiply. For example, to divide 14\frac{1}{4} by 32\frac{3}{2}, we need to invert the second fraction and multiply: 14Γ·32=14Γ—23=1Γ—24Γ—3=212=16\frac{1}{4} \div \frac{3}{2} = \frac{1}{4} \times \frac{2}{3} = \frac{1 \times 2}{4 \times 3} = \frac{2}{12} = \frac{1}{6}.

Q: How do we add fractions?

A: To add fractions, we need to have a common denominator. The least common multiple (LCM) of 2 and 6 is 6. We can rewrite the fractions with a common denominator: 32=3Γ—32Γ—3=96\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} and 16\frac{1}{6} remains the same. Now we can add the fractions: 96+16=106\frac{9}{6} + \frac{1}{6} = \frac{10}{6}.

Q: How do we simplify the result 106\frac{10}{6}?

A: The result 106\frac{10}{6} can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. 106=10Γ·26Γ·2=53\frac{10}{6} = \frac{10 \div 2}{6 \div 2} = \frac{5}{3}.

Q: What is the final result of the expression?

A: The final result of the expression is 53\frac{5}{3}.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

  • To evaluate a mixed number, convert it to an improper fraction.
  • To divide fractions, invert the second fraction and multiply.
  • To add fractions, have a common denominator and add the numerators.
  • To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

Q: What are some practice problems that can help reinforce the concepts learned in this problem?

A: Some practice problems that can help reinforce the concepts learned in this problem are:

  1. Evaluate the expression: 213+12Γ·342 \frac{1}{3} + \frac{1}{2} \div \frac{3}{4}
  2. Simplify the fraction: 1216\frac{12}{16}
  3. Add the fractions: 14+16+18\frac{1}{4} + \frac{1}{6} + \frac{1}{8}

Q: What are some solutions to the practice problems?

A: Some solutions to the practice problems are:

  1. 213+12Γ·34=73+23=93=32 \frac{1}{3} + \frac{1}{2} \div \frac{3}{4} = \frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3
  2. 1216=12Γ·416Γ·4=34\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}
  3. 14+16+18=624+424+324=1324\frac{1}{4} + \frac{1}{6} + \frac{1}{8} = \frac{6}{24} + \frac{4}{24} + \frac{3}{24} = \frac{13}{24}