What is 1 half of a whole?

Example 16. The percent that means a third.

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a) In a recent exam, a third of the class got A. What percent got A?

Answer. Since the whole class is 100%, then a third of the class will be a third of 100%. We must divide 100 by 3. It will not be a whole number. (Lesson 11.)

100
3

99 + 1
3

= 33 +

1
3

= 33

1
3

% of the class got A.

We see, then, that 33

1
3

% means a third.

Again, percents are parts of 100%. Just as 50% means half, because 50 is half of 100; and 25% means a quarter because 25 is a quarter of 100; so 33

% means a third. 33

is a third of 100.

b) What percent means two thirds?

Answer. Two thirds of 100 will be 2 ×

2 × 33

1
3

2 × 33

+ 2 ×

1
3

2 × 33

= 66 . 2 ×

1
3

2
3

% means two thirds.

In Section 2, Question 10, we will see a simple way to find a quarter or 25% of a number.

Percent continues in Lessson 17.

Example 17. Calculator problem. How much is five eighths of $650.16?

Solution. To find five eighths, we must first find one eighth, and then multiply by 5. Press

See:

Two theorems

We have seen that

Half of 100 + Half of 12 = Half of (100 + 12).

Here is the theorem:

1.	The sum of the same part of numbers is that same part of the sum of those numbers.

	Euclid, VII, 5.

Let the number A be a part of number C, and let number B be the same part of number D.

Then the sum of A and B will be that same part of the sum of C and D.

For since A is the same part of C that B is of D, there are as many numbers in C equal to A as there are in D equal to B.

Therefore divide C into the numbers equal to A, namely G, H, I,
and divide D into the numbers equal to B, namely J, K, L;

thus C and D have been divided into the same number of parts.

Then since G is equal to A, and J equal to B, the sum of G and J is equal to the sum of A and B.

For the same reason, the sum of H and K, and the sum of I and L, are also equal to the sum of A and B.

Therefore, as many numbers as there are in C equal to A, so many are there in the sum of C and D equal to the sum of A and B.

Therefore, whatever multiple C is of A, the sum of C and D is the same multiple of the sum of A and B.

Therefore, whatever part A is of C, the sum of A and B is the same part of the sum of C and D.

Which is what we wanted to prove.

That property of numbers is also true for parts, plural. For example:

Three fifths of 100 + Three fifths of 10 = Three fifths of (100 + 10).

Here is the theorem:

2.	The sum of the same parts of numbers is equal to the same parts of the sum of those numbers.

	Euclid, VII, 6.

Let the number A be the same parts of number C that number B is of number D.

Then the sum of A and B will be the same parts of the sum of C and D.

For since A is the same parts of C that B is of D, there are as many numbers in A equal to a part of C as there are in B equal a part of D.

Divide A into those numbers equal to a part of C, namely G, H, I. And divide B into those numbers equal to a part of D, namely J, K, L;

thus A and B have been divided into the same number of parts.

Then since G is the same part of C that J is of D, the sum of G and J is that same part of the sum of C and D. (Theorem 1.)

For the same reason, the sum of H and K, and the sum of I and L, are also that same part of the sum of C and D.

Therefore, whatever parts A is of C, the sum of A and B is the same parts of the sum of C and D.

Which is what we wanted to prove.

Since these are true theorems of arithmetic, then—when when multiplication by a fraction a is defined as in Lesson 27—the factoring axiom of algebra,

ab + ac = a(b + c),

can be applied to arithmetic.

At this point, please "turn" the page and do some Problems.

Continue on to the Section 2.

1st Lesson on Parts of Natural Numbers

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Welcome to the Smartick blog! In this week’s post, we are going to learn how to calculate halves, thirds, and fourths. These expressions are not only used in math problems, but also in daily life.

Do you know what they are? Do you know how to calculate them? In this post, you will realize how easy it is to calculate halves, thirds, and fourths.

Halves

One half is equivalent to the fraction: 1/2. Therefore, it is half of any amount. Halves are calculated by dividing by 2.

For example:

One half of 10 = ½ of 10 = 10/2 = 5.

One half of 34 = ½ of 34 = 34/2 = 17.

Three halves of 14 = 3/2 of 14 = 3 x 14/2 = 3 x 7 = 21.

Thirds

One third is equivalent to the fraction: 1/3. Therefore, it is a third of an amount. Thirds are calculated by dividing by 3.

For example:

One third of 24 =1/3 of 24 = 24/3 = 8.

One third of 33 =1/3 of 33 = 33/3 = 11.

Five thirds of 15 = 5/3 of 15 = 5 x 15/3 = 5 x 5 = 25.

Fourths

One fourth is equivalent to the fraction: 1/4. Therefore, it is a quarter of an amount. Fourths are calculated by dividing by 4.

For example:

One fourth of 20 = ¼ of 20 = 20/4 = 5.

One fourth of 28 = ¼ of 28 = 28/4 = 7.

Seven fourths of 8 = 7/4 of 8 = 7 x 8/4 = 7 x 2 = 14.

Have you learned about halves, thirds, and fourths? Feel free to share this post with your friends and colleagues so that they too can learn. And remember that in order to learn these calculations and much more it is best to sign up on Smartick and try it for free!

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