Solving the Mixed Number Division Problem

The problem asks us to find a number that, when used to divide the mixed number \(10\frac{2}{3}\), results in 20. We can represent this problem as an equation.

Let the unknown number be \(x\).

The equation can be written as:

First, we need to convert the mixed number \(10\frac{2}{3}\) into an improper fraction. To do this, multiply the whole number part (10) by the denominator (3) and add the numerator (2). Keep the original denominator (3).

Now substitute this improper fraction back into our equation:

Dividing by \(x\) is the same as multiplying by the reciprocal of \(x\), which is \(\frac{1}{x}\).

To solve for \(x\), we can multiply both sides of the equation by \(3x\) to eliminate the denominator on the left side:

Now, isolate \(x\) by dividing both sides of the equation by 60:

The resulting fraction \(\frac{32}{60}\) can be simplified. Find the greatest common divisor (GCD) of 32 and 60. Both numbers are divisible by 4.

Divide the numerator and the denominator by their GCD:

So, the number by which \(10\frac{2}{3}\) should be divided to obtain 20 is \(\frac{8}{15}\).

Let's check our answer:

Dividing by a fraction is the same as multiplying by its reciprocal:

We can cancel common factors. 32 and 8 have a common factor of 8. 3 and 15 have a common factor of 3.

The check confirms that our calculated value for \(x\) is correct.

Revision Table: Key Steps

Additional Information: Understanding Mixed Numbers and Division

A mixed number combines a whole number and a fraction, like \(10\frac{2}{3}\). To perform calculations easily, we often convert them to improper fractions, where the numerator is larger than the denominator. For \(10\frac{2}{3}\), this means \(\frac{32}{3}\).

Dividing by a number is the inverse operation of multiplying by that number. For example, dividing by 5 is the same as multiplying by \(\frac{1}{5}\). Similarly, dividing by a fraction, say \(\frac{a}{b}\), is equivalent to multiplying by its reciprocal, \(\frac{b}{a}\).

In this problem, we used the property that if \(A \div x = B\), then \(A = B \times x\), and therefore \(x = \frac{A}{B}\). We applied this with \(A = \frac{32}{3}\) and \(B = 20\), leading to \(x = \frac{32/3}{20}\). Calculating \(\frac{32/3}{20}\) is the same as \(\frac{32}{3} \div 20\), which is \(\frac{32}{3} \times \frac{1}{20} = \frac{32}{60} = \frac{8}{15}\). This provides an alternative way to solve the equation \(\frac{32}{3} \div x = 20\), by treating \(x\) as the divisor and rearranging the equation.