21. 4. 23. 8. 25. . 26. 18. 27. 우.

104. Divide as indicated, and carry out the remainder to three decimals:

SUMMARY OF FRACTIONS

105. 1. A fraction is one or more of the equal parts of a unit.

2. The numerator of a fraction expresses the number of parts taken.

3. The denominator shows into how many parts the unit has been divided. It, therefore, expresses the size of the parts.

4. Reduction is the process of changing the form without changing the value.

5. To reduce fractions to their lowest terms, we reject common factors from both numerator and denominator.

6. An improper fraction has a numerator greater than its denominator.

7. To change an improper fraction to an integer, mixed number, or decimal, we divide its numerator by its denominator.

8. A mixed number is a whole number and a fraction taken together.

9. To change a mixed number to an improper fraction, we change the whole number to the number of parts shown by the denominator of the fraction, then unite these parts with the fraction.

10. Similar fractions have like denominators. 11. Fractions must be made similar before they can be added or subtracted.

12. To add or subtract fractions, we change them to similar fractions, then find the sum or difference of their numerators, and write the result as the numerator of the fraction which has the common denominator.

13. To add or subtract mixed numbers, we add or subtract the fractional and integral parts separately.

14. When the least common denominator cannot be easily found without the process, we find the least common multiple of the denominators for the least common denominator.

15. To multiply fractions, we multiply the numerators together for the numerator of their product, and the denominators together for the denominator of their product.

16. In the multiplication of fractions we may often shorten the process by canceling equal factors from the numerators and the denominators. This process, called cancellation, does not change the value of the product.

17. To multiply an integer by a fraction, or a fraction by an integer, we may express the integer in fractional form, and apply the general rule for multiplication of fractions.

18. To divide by a fraction, we multiply by that fraction inverted.

19. In division of fractions, we may express the integers in fractional form, and apply the general rule.

20. To find the value of any fractional expression we divide the numerator by the denominator.

21. In using mixed numbers with large integral parts in multiplication, we may multiply the whole number and the fractional part separately, and then unite their products.

22. In dividing a mixed number with a large integral part by an integer, we divide the integral part as far as possible, then change the remainder, including the fractional part, to an improper fraction and complete the division.

23. To find what fractional part one number is of another, we express the number denoting the

part as the numerator, and the number denoting the whole as the denominator of a fraction.

24. To find the value of the whole unit when the value of a part is given, we divide the given value by the given part.

25. To change decimals to fractional form, we supply the indicated denominator and reduce the resulting fraction to its simplest form.

26. To change fractions to decimal form, we divide the numerator by the denominator.

27. Fractions indicate division. The value of a fraction is the quotient obtained by dividing the numerator by the denominator.

28. Both numerator and denominator of a fraction may be either multiplied or divided by the same number without changing the value of the fraction.

29. Multiplying the numerator of a fraction multiplies its value. Dividing the numerator of a fraction divides its value.

30. Multiplying the denominator of a fraction divides the value. Dividing the denominator of a fraction multiplies its value.

31. Any change in the numerator produces a like change in the value of the fraction.

32. Any change in the denominator produces an opposite change in the value of the fraction.

Solution of Problems

106. 1. Make an abstract or use a diagram to show clearly what is given and what is required in the problem.

2. Determine the general question of the problem, and the terms in which the answer must be expressed, that you may make the computations in abstract form.

3. Decide upon the steps of the solution, and indicate these steps clearly in the outline of your plan of solution. In the work of solution name clearly the results obtained by each step.

4. Perform the operations in the easiest way, the terms being arranged in the most convenient form for computation. Answer the questions step by step, giving full consideration to each one in order, as if it were a problem by itself.

5. Check or prove your answers.

6. It often helps, in making the outline, to substitute easy and simple numbers for the large or complex ones given, keeping the general questions the same as those of the given problem.

Miscellaneous Problems for General Review

107. 1. If of 9 bushels of wheat cost $131, what will ğ bushel cost?

2. If 7 tons of hay cost $120, how many tons can be bought for $78?