5 is equal to 4+1 is written 5=4+1; 6 is less than 8 is written 6<8; 7 is greater than 5 is written 7>5. Thus, = means "is equal to"; > means "is greater than"; < means "is less than." The pupil must make himself familiar with the meaning of these symbols. 13. Equation. A statement expressing the equality of two numbers, as 5=4+1, AB=3.24, is called an equation. EXERCISES Write each of the following statements in symbols: five is greater than three; eight is equal to the sum of six and two; seven is less than ten; AB is less than MN; x is greater than y; the difference between a and b is less than the sum of a and b. 14. Notation for line segments. We have denoted line segments by marking the end points with capital letters. Sometimes one small letter is used to denote a line segment. This is written on the segment near the mid- A point, as a (Fig. 38). a FIG. 38 Capital letters usually represent points, small letters denote numbers. In Fig. 38 the number a is an unknown number. It denotes the length of the segment, and may be found by measuring the segment. 15. Literal number. There are various ways of denoting numbers by symbols. For example, the number five may be denoted as follows: 5. V, , or . H. A number denoted by a letter is a literal number. COMPARING LENGTHS OF SEGMENTS BY MEANS Let 16. How to find the ratio of line segments. AB and CD be two line segments whose lengths are respectively 2.38 and 3.15, and let it be required to compare the length of AB with that of CD. have dropped all figures after the second decimal place. However, it is true approximately to the third figure, or to two decimal places. The quotient .75 is the ratio of AB to CD. This ratio compares segment AB with segment CD. It indicates that AB is about .75 of CD or as long as CD. In general, the ratio of two segments is the quotient found by dividing the measure of one by that of the other, provided a common unit is used in measuring. Thus, if one segment is 2 in. long and another 3 in., the ratio of the segments is 3, and the length of the first segment is of the length of the second. EXERCISES 1. Draw two segments, AB and CD. Measure the segments to three figures. Find the ratio by dividing one measure by the other. Arrange your work as shown in §16. State your results approximately to two decimal places. 2. If AB=3.16 and CD=1.24, find the ratio to three figures, arranging your work as in the example of §16. 3. In triangle ABC (Fig. 40) measure each of the segments AB, BC, and CA to three figures and find, to the third figure, the ratios a 23.8 by 24.1 1.40 by 915 0.12 by 218 B 17. How to find the ratio of two numbers. We have seen that lengths may be compared by means of ratios. Two numbers may be compared with each other by dividing. Thus, to compare 8 with 12 we divide 8 by 12. The comparison is then expressed by the ratio 2 or 3, meaning that 8 is of 12. 8 In the following table compare each number in the first line with the corresponding number in the second by dividing the number in the first by the corresponding number in the second. The quotient found by dividing 6 by 2 is the ratio of 6 to 2. In general, the ratio of a number to another is the quotient obtained by dividing. The ratios of 6 6 a 3 -" b to 3, and of a to b, may be written in the form and read "6 over 3," "a over b," meaning 6 divided by 3, a divided by b. In arithmetic is usually read "6 thirds." EXERCISES Express the ratios of the following, reducing each to the lowest terms. Dividing numerator and denominator first by 2 and then by 7, we have MEASURING BY HUNDREDTHS 18. Expressing length as hundredths of a unit. On squared paper draw a segment 10 cm. long (one decimeter). In the following this is to be used as a unit of measurement (Fig. 41). Imagine each centimeter divided into 10 equal parts (millimeters) as on the centimeter ruler. Each millimeter is of a centimeter, or 100 of the unit ( of a decimeter). 10 20 30 40 50 60 70 80 90 100 100 100 100 100 FIG. 41 100 100 100 Each centimeter is therefore 1 of the unit. 100 We shall now see how to measure line segments to the nearest one-hundredth part of the unit. EXERCISES 1. Show from a drawing that a segment equal to of a decimeter is equal to 10% of it. 2. Using a decimeter as a unit, express the following line segments as hundredths. 19. Meaning of per cent. In Exercise 2 hundredths were expressed by means of fractions having 100 as denominator. The words per cent, meaning hundredths, are also used. The sign %, read per cent, has the same meaning. EXERCISES 1. The results of Exercise 2 may be expressed as follows: ==20 hundredths = 20 per cent =20%. = 100 75 100 =75 hundredths=75 per cent=75%, etc. Express similarly the equivalents of 3, 3, 3, 3, 3, 3. 2 1 |