A Note on Ratios

Do you have two things to compare? If each can be summarized by a quantity, a ratio is quick way to compare them. This discussion answers the why's, the when's and the how's learners asks when first introduced to ratios.

THE CONTENT

1. Why a ratio?

2. How to represent a ratio

3. Comparing of unlike things

4. Miscellaneous examples

5. Ratio as fractions

6. A teaching moment

7. The arithmetic of ratios

8. Whole numbers and ratios

9. Teaching problem solving

10. A teaching device

11. Test understanding

12. An interesting ratio

1. Why a ratio?

A ratio is the number of times one quantity [1] is contained in the other. Thus it is easy to tell which quantity is the larger or the lesser.

2. How to represent a ratio

The ratio of the quantities, a and b, is expressed by the symbolism a : b. It is read a to b. This language is consistent with the everyday language of comparisons; this to that.

ain the symbol, a : b, is called the first term of the ratio and bis called second term of the ratio.

If the ratio is greater than one, the first term is associated with the greater sanity. If the ratio is equal to one the first and second term are associated with the same quantity. It the ratio is leas than one; the first term is associated with the lesser quantity.

3. Comparing unlike things is never permitted

Therefore the quantities used in a ratio must be of the same type. For example, a ratio cannot have one quantity in feet (a linear measure) and the other in ounces (a measure of weight or volume). Both quantities must be in feet or both must be in ounces.

4. Miscellaneous Examples

The Ratio of Forcible Rape to Violent Crime (2009)statistics: 88,097 : 1,318,398.

This is a ratio is less than one.

Ratio of 1955 assets: Exxon Mobil to GM:

6,614.7 to 5,130.1 (measured in millions of dollars)

This is a ratio is greater than one.

Ratio of S80 Turbo V6 engine to V70 5-cylinder engine: 2000 rpm: 2300 rpm.

The measurements were taken at 60 mph. The ratio is less than one, suggesting the S80 works less at that speed. Conclude the S80 power plant is more powerful. The test vehicles are the author's personal property so the results cannot be generalized.

This is as a simplistic measurement of these vehicles' power power plants.
5. Ratio as fractions

Ratios give the number of times one quantity contains the other. So do fractions. So a ratio can be expressed as a fraction.

a : b = a / b.

6. A teaching moment

Are you a teacher? Do you wish your students the opportunity to learn about their school and at the same time use ratios? Then assign the following problems.

a) Give the ratios of men teachers to women teachers in the school; b) What is the ratio of female math teachers to the male math teachers;

c) Give the ratio of boys to girls in your math class.

Of course you will have to supply the data.

7. The Arithmetic of Ratios

To Create Equal Ratios: Ratios are fractions. Multiplying the numerator and denominator of a fraction by the same positive number does not change the value of a fraction.

So, multiplying the first and second term of a ratio by the same positive number will not change value of the original the ratio.

In symbols, a : b = am bm , (where m is a positive number).

Thus 3 : 4 = 21 : 28. Why?

To Increase The Size Of A Ratio: Add the same positive number to each term of the ratio. If the beginning ratio is less than one, the new ratio will be less than one and it will be greater than the beginning ratio. [2]

In symbols: If a : b < 1, then for any positive number, m;

a : b < (a + m) : (b + m) < 1

3 : 4 < 8 : 9. Why?

8. Whole numbers and ratios

All ratios cannot be expressed with whole numbers.

For instance, the ratio of the length of the diagonal of a square to the length of a side of the square cannot be expressed with whole numbers. Nor is it possible to express the ratio of the length of a circle (circumference) to the length of its diameter with whole numbers.

These ratios can be approximate to any degree of accuracy.

Numbers that cannot be expressed as a ratio of whole numbers are called incommensurable.

9. Problem solving with worked examples

ILLUSTRATIVE EXAMPLES

1. Arrange these ratios in decreasing order:

7 : 9,

3 to 4,

16 : 41,

3 : 8,

2 : 7.

Express these ratios as fractions. Then arrange the fractions in decreasing order. The ratios will have the same ordering.

Use cross-multiplication to determine the order of fractions.

2. For what value(s) of x will the ratio (3 + x ): (4+x) be equal 5 to 6?

Express the ratio as a fraction then solve. (3 + x) / (4+x ) =5 / 6.

Get x = 2.

3. What number must be added to the terms of the ratio, 3 : 4, in order to get the ratio, 25 : 32?

Let x be the number that must be added to the numerator and denominator to get the ratio 25 : 32.

Using fractions, get (3+x) / (4+x) = 25 / 32.

Multiply the previous equation by 32(4+x).

Get 32(3+x)= 25(x+4).

Now solve to x = 4 / 7.

4. Find two numbers whose ratio is 5: 6 and their sum is 121?

Let x, y be the two numbers so their ratio is x : y

.

Use fractions for the ratios,

get x / y = 5 / 6 and x + y =121.

Multiply the last equation by 6y and

get: 5y=6x.

Substitute (121-x) for y in the equation on the prevuous line.

Get 5(121- x) = 6x.

Solve to get:x = 11*5 = 55.

So y = 121-55 = 66.

5. The ages of two people gives the ratio 3 : 4. Thirty years ago the ratio of their ages was 1 : 3. What are their present ages?

Let x : y be the ratio of their present ages.

Write x : y = 3 : 4 in fractions and get

x / y = 3 / 4. Multiply by 4x, the equation becomes 3x=4y.

Now thirty years ago their ratio was (x-30) / (y-30) = 1 / 3.

Simplify the last equation and get

3(x - 30) = y - 30 or 3x - 90 = y - 30

Multiply the last equation by 3.

9x - 270 = 3y - 90.

Replace the 3y in the last equation with 4x (see 6 lines above) get

9x - 270 =4x - 90 or 5x = 180

So x = 36 and y = 48.

6. Show that when x is positive and a > x then (a2 - x2) to (a²+x² )will be greater that a- x to a+x.

Attack: Change the ratios to fractions. Thenexpress the first fraction as a product of the second fraction and a number called, F.

If the size of F > 1, the first fraction/ratio is the greater.

If the size of F = 1, the fractions/ratios are equal.

If the size of F < 1, the first fraction/ratio is the smaller.

Factor the numerator of the first fraction. Then multiply the numerator and denominator by the positive number (a+x) and get:

(a²-x² ) / (a²+x² ) = (a-x)(a+x) (a+x) / (a+x ) (a²+x²) =

(a-x)/(a+x) * [ (a+x)² / (a²+x²) ]

F is the fraction in the brackets. Since (a+x)² > ( a²+x² ), F >1.

Therefore the first ratio is the greater.

7. Find x : y, if x² + 15y² = 2xy.

Attack: Divide the equation by y^{2. .} Then arrange to

(x/y)² - 2(x/y) + 15 = 0.

Factor this quadratic. Get, x/y =5.

8. What is x to y, if (4x + 5y)/(3x - y) = 2 ?

Attack: Multiply the equation by (3x-y).

Get (4x+5y)= 2(3x-y). Solve to

2 x = 7 y .

Divide the equation by 7y.

x / y = 7 / 2 , the desired ratio.

10. A teaching device

Having delivered an excellent explanation of a new concept and now certain the learner has mastered the language of the concept, problem solving skills are quickly grasped after learners have studied a set of worked examples that illustrate the problem solving techniques.

Try it. Below are ILLUSTRATIVE EXAMPLES for learnersto study; alone or in groups. Inform them, they will be tested on these examples before the end of the class period. Then test them. Experiment with the amount of study time given the student. Start with 20 minutes.

During the following class meeting, test learners on the same problems. But this time, change the variable names. Keep the greater of the two grades.

11. Test understanding

Use these EXTRA PROBLEMS to determine the level of the learners' understanding on solving ratio problems.

EXTRA PROBLEMS

1. For what value of x will (14+x) : (16 + x) equal 8 : 9?

2. Find x : y; if x and y satisfy 9x² +y² = 6xy.

3. Arrange in ascending order:

3 ; 4

1 : 2 ,

13: 19,

5 : 8,

13 : 14

4. Find x : y ; if (4x+5y)/(3x-y) = 4.

5. Show that, if from each term of a ratio the multiplicative inverse of the other term is subtracted, the new ratio is the original ratio. Multiplicative inverse of the number, w is 1 / w.

6. The ages of two persons have ratio 3 : 4. Thirty years ago the ratio of their ages was 1: 3, what are their present ages?

[1] Quantity is a basic property of a thing. It is measurable. Quantities can be divisible or indivisible. Divisible quantities are a collection of objects like a flock, or a herd, a group of people or a company of soldiers. When a quantity is indivisible, it refers to something continuous; for instance liquid, heat, or time. Quantities expressed by numbers can be ranked by size.

[2] Treat the ratios as fractions. Subtract both ratios from one. Note they have the same numerator.

1 - a / b = (b-a) / b

1 - (a+m) / (b+m) = (b-a) / (b+m)

But the denominator of the second difference is greater than the denominator of the first difference. This means it is the lesser of the two differences. Hence the ratio of this subtraction is closer to one than the other. It makes it larger since both ratios are less than one. So the second difference is a smaller number. Hence the second ratio is closer to one than the first ratio.

12. An Interesting Ratio

On the line segment AB, let C be the mean proportional between the points A and B. The ratio BC/AB is incommensurate . The ancients knew this ratio. They used it in their architecture, to describe beauty and balance. Today it is used in modern art design. It's helpful in the explanations of many phenomena, as well as in designs in the human body.

A simple procedure shows the ratio value is between .6175 and .6182. Many presentations approximating this value exist. A simple one using only basic rules of fractions and elementary terminology of line segments is provided here.

This ratio reveals an interesting aspect of mathematics. It casts light on our understanding of many things, even though it was not studied for this information. This is not uncommon for a mathematical study to reveal useful information that was not the target of its creation. This reality alone gives mathematics its power and beauty.

12. Challenging Problem

Take C to be the mean proportional between the points, A and B.

Compute the 4^th approximate to the ratio AC : AB. Use simple continued fraction techniques.

G. Chrysral, Textbook of Algebra Volume II 7^th Edition Chelsea Publishing Company

Charles Smith, Elementary Algebra, The Macmillian Company