« AnteriorContinuar »
1. It is usual to say that we have measured any concrete quantity, when we have found out how many times it contains some familiar quantity of the same kind.
We say for example, that we have measured a line, when we have found out how many feet it contains. We say that we have measured a field, when we have found out how many acres or how many square yards it contains.
To know the measurement of any quantity then, we must have two things. First, we must have a unit, or standard of reference, of the same kind as the thing measured. Secondly, we must have the measure, or the number of times the thing measured contains the unit, or standard quantity.
3. Hence, the measure of a quantity is the number, and the unit is the concrete quantity, by means of which it is measured.
Example 1. A line contains 261 feet. Here the measure or number is 261 and the unit a foot.
Example 2. What is the measure of 24 miles when a yard is the unit?
2} miles=5 x 1760 yards,
=4400 yards=4400 x 1 yard, therefore the measure is 4400 when a yard is the unit.
Example 3. What is the unit when the measure of a field of 10 acres is 242 ?
Example 4. If the unit be a yards, what is the measure of b miles ?
b miles=b x 1760 yards,
b x 1760
(1) What is the measure of 1 mile when a chain of 66 feet is the unit?
(2) What is the measure of an acre when a square whose side is 22 yards is the unit ?
(3) What is the measure of a ton when a weight of 10 stone is the unit ?
(4) The length of an Atlantic cable is 2300 miles and the length of the cable from England to France is 21 miles. Express the length of the first in terms of the second as unit.
(5) The measure of a certain field is 22 and the unit 1100 square yards : express the area of the field in acres.
(6) Find the measure of a miles when b yards is the unit.
(7) The measure of a certain distance is a when the unit is c feet. Express the distance in yards.
(8) A certain sum of money has for its measures 24, 240, 960 when three different coins are units respectively. If the first coin is half a sovereign, what are the others ?
4. The measure of a quantity is the number of times which that quantity contains the unit. We may express the same thing in different ways,
(i) The measure of a quantity is the ratio of that quantity to the unit.
(ii) The measure of a quantity is the fraction that the quantity is of the unit.
5. This last statement is in the language of Arithmetic, and the word 'fraction' is to include whole numbers, and improper fractions.
6. The following are therefore different forms of the same question :
(i) What is the measure of 4 miles when 66 feet is the unit? (ii) How many
times does 4 miles contain 66 feet? (iii) What is the ratio of 4 miles to 66 feet?
(iv) What fraction of 66 feet is 4 miles ? Example (i) What is the measure of a yards when 6 feet is the unit? a yards=3a feet x 6 feet,
x b feet. b
За .. the required ratio is
(1) The ratio of the area of one field to that of another is 20 : 1, and the area of the first is half a square mile. Find the number of square yards in the second.
(2) The ratio of the heights of two persons is 9 : 8, and the height of the second is 5 ft. 4 in. What is the height of the first?
(3) The measure of a field with 3 acres for unit is 65. Find the ratio of the field to an acre.
(4) One field contains a second of 2} acres, 64 times.
(7) What is the measure of a miles when 6 chains is the unit? How many times do a miles contain c chains ?
hat is the ratio of a miles d chains ? What fraction is a miles of k chains ?