Calculating the Total Length of a Big Rod by Joining Small Rods

This problem asks us to find the total length of a big rod created by joining several smaller rods end-to-end. We are given the number of small rods and the length of each individual small rod. To find the total length, we need to multiply the length of one small rod by the total number of small rods.

Let's break down the given information:

• Number of small rods = 15
• Length of each small rod = \(23\frac{2}{7}\) meters (m)

The length of each small rod is given as a mixed number. To perform multiplication easily, it's best to convert this mixed number into an improper fraction.

Converting Mixed Number to Improper Fraction

A mixed number like \(a\frac{b}{c}\) can be converted to an improper fraction using the formula \(\frac{(a \times c) + b}{c}\).

For the length \(23\frac{2}{7}\) m:

Here, \(a = 23\), \(b = 2\), and \(c = 7\).

Improper fraction = \(\frac{(23 \times 7) + 2}{7}\)

Calculating the numerator:

\(23 \times 7 = 161\)

\(161 + 2 = 163\)

So, the improper fraction is \(\frac{163}{7}\).

The length of each small rod is \(\frac{163}{7}\) m.

Calculating the Total Length of the Big Rod

The total length of the big rod is the product of the number of small rods and the length of one small rod.

Total Length = Number of rods \(\times\) Length of each rod

Total Length = \(15 \times \frac{163}{7}\)

To multiply a whole number by a fraction, we multiply the whole number by the numerator of the fraction and keep the denominator the same.

Total Length = \(\frac{15 \times 163}{7}\)

Let's calculate the product \(15 \times 163\):

\(163 \times 15 = 2445\)

So, the total length in meters is \(\frac{2445}{7}\).

Converting Improper Fraction back to Mixed Number

The total length is \(\frac{2445}{7}\) m. To express this as a mixed number, we need to divide the numerator (2445) by the denominator (7).

We perform long division:

\(2445 \div 7\)

• Divide 24 by 7: \(24 = 7 \times 3 + 3\). Quotient is 3, remainder is 3.
• Bring down the next digit (4) to make 34.
• Divide 34 by 7: \(34 = 7 \times 4 + 6\). Quotient is 4, remainder is 6.
• Bring down the next digit (5) to make 65.
• Divide 65 by 7: \(65 = 7 \times 9 + 2\). Quotient is 9, remainder is 2.

The quotient of the division is 349, and the remainder is 2. The denominator remains 7.

So, the improper fraction \(\frac{2445}{7}\) is equal to the mixed number \(349\frac{2}{7}\).

The total length of the big rod is \(349\frac{2}{7}\) m.

Conclusion

By joining 15 small rods, each \(23\frac{2}{7}\) m long, the total length of the resulting big rod is \(349\frac{2}{7}\) m.

Revision Table: Key Steps in Calculating Total Length

Additional Information: Working with Fractions

Understanding how to work with fractions, especially converting between mixed numbers and improper fractions, and performing operations like multiplication, is crucial for solving many mathematical problems.

• Mixed Numbers: A mixed number combines a whole number and a proper fraction (where the numerator is less than the denominator).
• Improper Fractions: An improper fraction has a numerator that is greater than or equal to its denominator.
• Multiplication of Fractions: To multiply fractions, multiply the numerators together and multiply the denominators together. When multiplying a whole number by a fraction, treat the whole number as a fraction with a denominator of 1 (e.g., \(15 = \frac{15}{1}\)).
• Units: Always remember to include the appropriate units (like meters in this case) in your final answer when dealing with physical quantities like length.

This problem is a good example of applying fraction arithmetic to solve a practical length calculation question.