12. Forming proportions from equations. Since proportions are algebraic equations, they may be rearranged in accordance with the laws of algebra. For example, if

x =

ab —— c

,

(1)

we may write the proportion

x — 1

=

ab —— c

,

(2)

or we may divide both sides by a to get

x — a

=

ab —— ac

,   or

x —— a

=

b — c

,

(3)

or we may multiply both sides by c/x to obtain

cx —— x

=

cab ——— xc

,   or

c —— 1

=

ab —— x

.

(4)

Rule (A). A number may be divided by 1 to form a ratio. This was done in obtaining proportion (2). Rule (B). A factor of the numerator of either ratio of a proportion may be replaced by 1 and written as a factor of the denominator of the other ratio, and a factor of the denominator of either ratio may be replaced by 1 and written as a factor of the numerator of the other ratio. Thus (3) could have been obtained from (1) by transferring a from the numerator of the right hand ratio to the denominator of the left hand ratio.

For example, to find

16 × 28 —————— 35

,    write x =

16 × 28 —————— 35

, apply Rule (B) to obtain

C — D

:

x —— 16

=

28 ——— 35

,

and                               push hairline to 35 on D,                               draw 28 of C under the hairline;                               opposite 16 on D, read x = 12.8 on C. Figure 3 indicates the setting. FIG. 3. To recall the rule for dividing a given number M  by a second given number N,

write x =

M —— N

, apply Rule (A) to obtain

D —— C

:

x —— 1

=

M —— N

,

and                               push hairline to M on D,                               draw N of C under the hairline;                               opposite index of C, read x on D.

To recall the rule for multiplication, set x =

MN ——— 1

, apply Rule (B) to obtain

D — C

:

x —— M

=

N ——— 1

,

and                               to N on D set index of C;                               opposite M on C, read x on D.

To find x if

1 ——— x

=

864 ————————— (7.48) (25.5)

, use Rule (B) to get

7.48 ——— x

=

864 ——— 25.5

, make the corresponding setting and

read x = 0.221. The position of the decimal point was determined by observing that x must be about 1/40 of 8, or 0.2. EXERCISES Find in each case the value of the unknown quantity.

1.   y  =   8 × 12 ————— 7 .

8.   498  =   89.3x ————— 0.5631 .

2.   7.4  =   9y —— 28 .

9.   0.874  =   3.95 × 0.707 —————————— x .

3.   8y  =   75.6 × 9    .

10.   0.695  =   0.0879 —————— x .

4.   y  =   86 × 70.8 ——————— 125 .

11.  1 ——— 386  =   0.772 ———— 2.85y .

5.   y  =   147.5 × 8.76 —————————— 3260 .

12.   2580y  =  17.9 × 587 .

6.   y  =   0.797 × 5.96 —————————— 0.502 .

13.   3.14y  =  0.785 × 38.7 .

7.  37 × 86 ——————— y   =   75.7 .

14.  0.876y ————— 5.49  =  7.59 .