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An improper fraction is a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This means the fraction represents a quantity greater than or equal to one.

For example, in the fraction \(\frac{7}{4}\), the numerator 7 is greater than the denominator 4.

To convert a mixed number to an improper fraction, you should:

• Multiply the whole number part by the denominator of the fractional part.
• Add this result to the numerator of the fractional part.
• Place this sum over the original denominator.

For instance, converting the mixed number \(\frac{1 \frac{3}{4}}\) to an improper fraction involves multiplying the whole number 1 by the denominator 4, which gives 4, then adding the numerator 3, to get 7. Thus, \(\frac{1 \frac{3}{4}} = \frac{7}{4}\).

A mixed number consists of a whole number and a proper fraction. For example, \(\frac{1 \frac{3}{4}}\) is a mixed number.

To convert an improper fraction back to a mixed number:

• Divide the numerator by the denominator to get the whole number part.
• The remainder becomes the numerator of the fractional part.
• The denominator remains the same.

For instance, converting \(\frac{7}{4}\) back to a mixed number involves dividing 7 by 4 to get 1 with a remainder of 3. Thus, \(\frac{7}{4} = 1 \frac{3}{4}\).

Equivalent fractions represent the same value, even though they may look different. You can find equivalent fractions by multiplying or dividing the numerator and the denominator by the same number.

For example, the fractions \(\frac{2}{4}\) and \(\frac{1}{2}\) are equivalent because \(\frac{2}{4} \big(\frac{2 \div 2}{4 \div 2}\big)= \frac{1}{2}\).

In the original exercise, both \(\frac{7}{12}\) and \(\frac{7}{4 \times 3}\) are simplified to the same equivalent fraction, \(\frac{7}{12}\), proving that the ratios form a proportion. To recognize equivalent fractions, always simplify the fractions and compare them.
If they reduce to the same simplified fraction, they are equivalent and hence form a proportion.