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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 2 miles when a yard is the unit?
2 miles = × 1760 yards,
=4400 yards=4400 × 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
Example 4. If the unit be a yards, what is the measure of
(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 6 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 :
What is the measure of 4 miles when 66 feet is
(ii) How many times does 4 miles contain 66 feet?
Example (i) What is the measure of a yards when b feet is the unit?
(ii) How many times does a yards contain b feet?
As in (i), a yards= xb feet.
(iii) What is the ratio of a yards to b feet?
(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, 63 times.
What is the measure of the first field in terms of the second ?
What is the ratio of the first field to the second?
Express the first as the fraction of the second.
(5) A certain weight is 3.125 of a ton.
What fraction is it of 4 cwt. ?
The ratio of a certain sum of money to 3 guineas is 22. Find its measure in terms of one pound.
Find the unit when its measure is 22.
(7) What is the measure of a miles when b chains is the unit?
How many times do a miles contain c chains?
What is the ratio of a miles to d chains?
What fraction is a miles of k chains?
What is that sum ?
(8) What is the unit when the measure of £20 is 400? £25 contains a certain sum 500 times. The ratio of £30 to a certain sum is 600. What is that sum? The fraction which £10 is of a certain sum is 200. What is that sum?
7. It is explained in Arithmetic, in the application of square measure, that the measure of the area of a rectangle is found in terms of a square unit, by multiplying together the measures of the sides in terms of the corresponding linear unit.
Example. Find in square feet the measure of a square
surface whose side is 12 feet.
The area is 12 × 12 square feet = 144 × 1 square foot, .. the measure required is 144.
8. We shall apply this result to Euclid I. 47.
Example 1. The sides containing the right angle of a rightangled triangle are 3 ft. and 4 ft. respectively; find the length of the hypotenuse.
Let a be the number of feet in the hypotenuse.
Then by Euclid I. 47, the square described on the side of x feet=the sum of the squares described on the sides of 3 feet and 4 feet respectively,