How To Find What Percent A Number Is Of Another

Lesson 30

WHAT PERCENT?

The Method of Proportions

The first lesson on percent is Lesson four.

In this Lesson nosotros hope to requite the student an understanding rather than only a mechanical method to get an answer.

In Lesson fourteen we saw how to use a calculator to solve percent problems. And in Lesson 18 we first introduced the method of proportions. In this Lesson we will see problems that anyone who understands percent tin easily do mentally. Arithmetics is and e'er has been a spoken skill.

To understand how we apply the terms Amount and Base of operations, see Lesson 28.

1.	Any number is what percent of 100?
5 is  ?% of 100.
	Any number is that percent of 100.

v is five% of 100.  12 is 12% of 100.  250 is 250% of 100.

For, a percentage is a number of hundredths.  5 is 5 hundredths -- v% -- of 100. That is the ratio of 5 to 100.  For a pct expresses a ratio, a relationship, between ii numbers.  What ratio has 6 to 12?  6 is half of 12, or, in the linguistic communication of percent, 6 is 50% of 12.

When the percent is less than or equal to 100%, then we can say "out of" 100.  25% is 25 out of 100.  But 250% cannot mean 250 out of 100 -- that makes no sense.  It means 250 for each 100, which is two and a half times. (Lesson xvi.)

Example 1.   $42.10 is what pct of $42.10?

Answer.   100%.  100% is all.

The method of proportions

24 out of 100 is 24%.  Equivalently, 24 is 24% of 100.

Just what percent is 24 out of 200?  24 is what percent of 200?

Answer. Per centum is how many out of 100.  But if there are 24 out of 200 --

-- then out of each 100 there are half every bit many:12.

24 is 12% of 200.

Since percents are ratios, then every bit a proportion problem, nosotros must find the missing term:

24 out of 200  is equal to____ out of 100?

When nosotros say "24 out of 200," then the Base -- the number that follows "of" -- is 200.  But with percentage, the Base of operations must be 100.  What must we exercise then to 200 to arrive 100?  We must take one-half of it or, equivalently, divide it past 2.  Therefore we must also divide 24 by ii.

24 out of 200  is equal to12 out of 100.

But 12 out of 100 is 12%.  Therefore, 24 out of 200 is also 12%.

Example 3.   What percentage is 24 out of 50?  Equivalently, 24 is what percent of l?

Reply.   Again, percent is how many out of 100.  Only if at that place are 24 out of 50 --

-- so out of 100, at that place are 2 24'southward:  48.

24 is 48% of l.

Proportionally,

24 out of 50 is equal to48 out of 100.

At present without even having to write that, the educatee should see that to go from base 50 to base 100, we accept to multiply by ii.  Therefore nosotros must multiply 24 past 2, also.

An all likewise common method these days is to make this an algebra problem.

The educatee is taught to cross-multiply and solve for x. That is a method for someone who does non understand percent. Whoever does understand percent will either practice such a trouble mentally, or, if that is also difficult, with a calculator: Percent = Corporeality ÷ Base.

ii.	How do we notice the Pct by the method of proportions?
	Solve the proportion in which 100 is the fourth term, and the Amount and Base are the first and second.
"The Amount is to the Base  aswhat number is to 100?"

The Base volition ever follow "of."  Therefore to find what percentage one number is of some other, we must make the Base 100.

(Simply see below.)

Example 4.   3 out of 10 students got A.  What percent got A?

That is, three is what percent of 10?  Those questions mean the same. In each 1, the base x follows "of."

Solution.  On multiplying both numbers past 10,

3 out of 10  is equal to  30 out of 100.

30% got A.

Example five.   In a grade of 25 students, xi studied French.  What per centum studied French?

Solution.   The pupil should understand that this means,

11 out of 25 studied French.

To make the base of operations 100, nosotros must multiply 25 by 4.  Therefore we must too multiply 11 by 4:

11 is to 25  as44 is to 100.

44% studied French.

While information technology is possible to practice this problem past writing the proper fraction , so solving this proportion:

the student should not have to write that. On seeing, or even hearing, the phrase "xi out of 25," the student should understand that nosotros must multiply both eleven and 25 by 4.

Understanding that is skill in arithmetic.

Example six.   In a grade of 200 students, 11 studied French.  What percent studied French?

Solution.  In this case, to brand 200 into 100, we must dissever by 2, or take half.  Therefore we must also accept half of xi, which is v½.

5½% studied French.

Nosotros see that to solve a proportion:

We must divide both terms past the aforementioned number,
or we must multiply both terms by the same number.

Case 7.   What percent of 400 is 33?

Solution.  400 is the Base; information technology follows "of."  To go far 100, nosotros must divide past 4.  Therefore we must also divide 33 past 4.

"4 goes into 33 8 (eight) times (32) with ane left over."

33 is 8¼% of 400.

Example 8.   g people voted in the recent ballot, and 763 voted for Jones.  What percent voted for Jones?

Solution.  763 out of 1000 voted for Jones.  To make the Base of operations 100, we must divide by 10.  To divide 763 by ten, but split one decimal digit.

763 ÷ 10 = 76.3

76.3% voted for Jones.

Example 9.   What percent of 500 is 132?

Solution.   132 is to 500  as  264 is to 1000  as  26.four is to 100.

132 is 26.iv% of 500.

Example 10.   7 out of 12 people voted Yes.  Approximately what percentage voted Yes?

Solution.   What number times 12 is shut to 100?

8 × 12 = 96.8 × seven = 56. Therefore approximately 56% voted Yes.

The ratio of the Corporeality to the Base

Example 11. 8 is what pct of 40?

Answer.  That question means the same equally, "8 has what ratio to 40?" For percents are ratios.

Now, viii is the fifth part of 40.  And since 20 is the fifth part of 100:

viii is 20% of 40.

To master the field of study of percent, the student should master Trouble i of Lesson 28.

Example 12.   What percent of 8 is 40?

Answer.   Hither, 8 is the Base -- it follows "of."  Information technology is possible to see that better if we express the question in standard grade:

xl is what percent of 8?

Now, 40 is five times 8.  The percent will have that same ratio to 100%. Five times 100% is 500%.

xl is 500% of 8.

Example 13.   In a class of 28 students, 7 got A.  What per centum of the class got A?

Answer.   The question is:  seven is what percentage of 28?

It is not obvious how to make 28 into 100.  But 7 is the fourth office, or a quarter, of 28.  And every bit a percent, a quarter of a number is 25%.

7 is 25% of 28.

Example 14.   12 is what percent of 18?

Answer.   12 has what ratio to xviii?  They have a common divisor vi.

"six goes into 12 two times and into 18 three times."

12 is to 18  equally2 is to three --

12 is 2 thirds of 18.

Each number says its proper name.

As a percent, 66% means two thirds.

12 is 66% of 18.

Example 15.   What percent of 35 is 14?

Answer.   In standard form,

fourteen is what per centum of 35?

They take a common divisor vii.  Therefore, on dividing each term by 7:

fourteen is to 35  astwo is to 5.

2 is two fifths of 5.  Two fifths of 100% is forty%. (I fifth is xx%.)  Therefore,

14 is twoscore% of 35.

Example xvi.   20 is what per centum of 8?

Answer.   20 is fabricated up to two 8'southward -- 16 -- plus 4, which is half of 8. 20 is ii and a half times viii.

xx is 250% of viii.

Case 17.  Immigration of decimals..eight is what percent of four?

Reply.   To solve any percentage problem, the numbers must exist whole numbers.  Therefore, make the numbers whole numbers by multiplying by ten:

.8 is to 4  every bit  8 is to 40.

At present, 8 is the 5th part of 40.  That tells united states of america,

.8 is 20% of 4.

Case 18.   What percent of 75 is 21?

Solution 1. In standard form,

21 is what per centum of 75?

They have a mutual divisor 3. That tells us that each number has a 3rd role.  And on taking a third of each one:

21 is to 75  every bit7 is to 25.

We can now modify 25 to 100 past multiplying past 4. Therefore, on as well multiplying 7 by 4:

21 is 28% of 75.

Solution 2.

21 75

=

? 100

What number times 75 volition produce 100?  In other words, What ratio has 100 to 75?

Answer:

100 is one and a third times 75.

Lesson xviii

One times 75 is 75.One tertiary of 75 is 25.  75 plus 25 is 100.

Therefore, one and a third times 21 is 21 + 7 = 28.

21 is 28% of 75.

Example 19.  Same digits.   $two.50 is what pct of $250?

Respond.  The digits are the same:  2, 5, 0; but $ii.50 has two decimal digits.  Information technology is $250 divided by 100:

$2.50 = $250 ÷ 100.

Therefore,

$ii.50 is 1% of $250.

Lesson iv.

Compare Lesson 15, Examples five and half-dozen.

Case twenty.   $84.l is what percent of $845?

Answer.   x%.  $84.50 = $845 ÷ 10.

Which is how to take 10% of a number.

Summary

To find what percent i number is of another:

Multiply or divide both terms then that the Base becomes 100.

9 is what percent of 20?

It is easy to brand 20 into 100 by multiplying information technology past v.  And since v times 9 is 45,

nine is 45% of 20.

When information technology is not obvious how to make the base 100: Look directly at the ratio of the Amount to the Base of operations.

7 is what percent of 28?

Since 7 is one quarter of 28, and so

vii is 25% of 28.

Example 21.   742 people were surveyed, and 213 responded No.  What percent responded No?

Solution.   213 is what per centum of 742?  Use your calculator

At this point, please "plow" the folio and do some Problems.

or

Proceed on to Section 2: A general method for finding the Percentage.

1st Lesson on Percent

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