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111. TO SUBTRACT SEVERAL FRACTIONS FROM THE SUM OF SEVERAL OTHERS.

Rule-1. Add together those which are to be subtracted, and the others in two separate sums.

2. Proceed to subtract by the previous rule.

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D.-Multiplication of a Fraction by an integer.

Rule-1. If the fraction be a mixed number, reduce it to an improper fraction.

2. Form a fraction, whose numerator is the product of the numerator and the multiplier; and the denominator that of the multiplicand.

3. Cancel any factors common to numerator and denominator; the resulting fraction is the product required.

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E.-Division of a Fraction by an integer.

Rule-1. If the fraction be a mixed number, reduce it to an improper fraction.

2. Form a fraction, whose numerator is the numerator of the dividend, and the denominator the product of the divisor and the denominator of the dividend.

3. Cancel any factors common to numerator and denominator, the resulting fraction is the quotient required.

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F.--Reduction of a Compound Fraction to a Simple Fraction.

Rule-1. Form a fraction, whose numerator is the product of all the numerators of the simple fractions, and whose denominator is the product of all the denominators. At first only indicate these products by writing the sign of multiplication between the factors.

2. Cancel all factors common to numerator and denominator.

3. Multiply all the remaining factors. The resulting fraction is that required.

4. If any of the simple fractions be mixed numbers, they must be reduced to improper fractions before applying the Rule.

EXAMPLES.

1. Reduce of 3 of of to a simple fraction.

2.

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Reduce 4 of of 83 of 23 to a simple fraction.

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Rule-1. Reduce mixed numbers, if there be any, to improper fractions. 2. Form a fraction, whose numerator is the product of all the numerators, and denominator the product of all the denominators; at first indicating only the multiplication by the proper sign.

3. Cancel all factors common to numerator and denominator.

4. Multiply together all the remaining factors, the resulting fraction is the product required.

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E.-Division of a Fraction by an integer.

Rule-1. If the fraction be a mixed number, reduce it to an improper fraction.

2. Form a fraction, whose numerator is the numerator of the dividend, and the denominator the product of the divisor and the denominator of the dividend.

3. Cancel any factors common to numerator and denominator, the resulting fraction is the quotient required.

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F.-Reduction of a Compound Fraction to a Simple Fraction.

Rule-1. Form a fraction, whose numerator is the product of all the numerators of the simple fractions, and whose denominator is the product of all the denominators. At first only indicate these products by writing the sign of multiplication between the factors.

2. Cancel all factors common to numerator and denominator.

3. Multiply all the remaining factors. The resulting fraction is that required.

4. If any of the simple fractions be mixed numbers, they must be reduced to improper fractions before applying the Rule.

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