Transcript Ratios - freshmanreview

We use ratios to make comparisons between
two things.
When we express ratios in words, we use the
word "to" –
we say "the ratio of something to something
else" Ratios can be written in several different
ways.
Multiplying or dividing each term by the same
nonzero number will give an equal ratio.
For example, the ratio 2:4 is equal to the ratio 1:2.
To tell if two ratios are equal, use a calculator and
divide.
If the division gives the same answer for both
ratios, then they are equal.
How can we write the ratio of squares
to circles, or 3 to 6?
The most common way to write a ratio
is as a fraction, 3/6.
We could also write it using the word
"to," as "3 to 6.“
Finally, we could write this ratio
using a colon between the two
numbers, 3:6.
Be sure you understand that these
are all ways to write the same
number.
ratio of squares to
circles is 3/6
ratio of squares to
circles is 3 to 6
ratio of squares to
circles is 3:6
Some other equal ratios:
3:6 = 12:24 = 6:12 = 15:30
Are 3:12 and 36:72 equal ratios?
Find 3÷12 = 0.25 and 36÷72 = 0.5
The quotients are not equal —> the ratios are not
equal
You can also use decimals and percents to
compare two quantities.
In our example of squares to circles,
we could say that the number of squares is "fivetenths" of the number of circles, or 50%.
EXAMPLE 1: REDUCE 36 : 72 TO THE SIMPLEST FORM.
SOLUTION:
GIVEN: RATIO 36 : 72.
TO FIND: SIMPLEST FORM OF RATIO.
HCF OF 36 AND 72 IS 36.
36 : 72 = 36/72 (DIVIDE EACH TERM BY HCF I.E. 36)
= (36/36)/(72/36) (SIMPLIFY)
= 1/2
=1:2
ANSWER: RATIO 36 : 72 IN SIMPLEST FORM IS 1 : 2.
Example 2: Find the simplest form of ratio of
1.5kg to 500g.
Solution:
Given: 1.5kg to 600g.
To find: Simplest form of ratio.
Ratio of 1.5kg to 600g = 1.5kg/600g (convert kg
to grams).**
= (1.5 x 1000)g/600g.
= 1500/600 (divide both terms by HCF i.e. 300).
= 5/2.
= 5 : 2.
Answer: Required ratio is 5 : 2.
Example 3: Which ratio is greater 2: 3 or 5 : 7.
Solution:
Given ratios 2 : 3 and 5 : 7.
To find: Which ratio is greater.
First ratio 2 : 3 = 2/3.
Second ratio 5 : 7 = 5/7.
LCM of denominators 3 and 7 is 21.
Multiply both ratios by 21.
Therefore,
First ratio = 2/3 x 21 (simplify).
= 2 x 7.
= 14.
Second ratio = 5/7 x 21 (simplify).
= 5 x 3.
= 15.
Since 15 > 14, Ratio 5 : 7 > 2 : 3
Example 4: Are the ratios 3 to 4 and 6 : 8 equal?
Solution:
Given: Ratios 3 : 4 and 6 : 8
To find: Are they equal.
Method I:
First ratio 3 : 4 = ¾.
Second ratio 6 : 8 = 6/8 (divide both terms by HCF i.e. 2).
= 3/4
Each ratio is 3/4. Therefore, the ratios are equal.
Answer: The ratios are equal.
Method II:
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are equal.
Therefore,
3 × 8 = 4 × 6.
24 = 24.
Since both of these products are equal to 24, the ratios are equal.
Example 5: Simplify the ratio: 2/3 : 4/7 : 3.
Solution:
Given: 2/3 : 4/7 : 3.
To find simplest form
Ratio 2/3 : 4/7 : 3 = 2/3 : 4/7 : 3/1**.
LCM of denominators 3, 7, and 1 is 21.
Multiply each term of the given ratio by 21.
Therefore,
Ratio 2/3 : 4/7 : 3/1 = (2/3 x 21) : (4/7 x 21) : (3/1
x 21).
= (2 x 7) : (4 x 3) : (3 x 21)
= 14 : 12 : 63
Answer: Simplified ratio is 14 : 12 : 63.
**While solving fractions you should write
whole numbers in fraction form dividing it by 1
e. g. 3 (whole number) = 3/1 (fraction).
Example 6: Joliet has a bag with 8
mangoes, 14 apples, 12 oranges and 1
pine-apple. What is the ratio of mangoes to
oranges?
Solution:
Step 1: Write what is given and what is to
be found.
Given: Mangoes 8, apples 14, oranges 12
and pineapple 1.
To find: Ratio between mangoes and
oranges.
Step 2: Write the expression.
(Hint: Write the numerator
equal to the first quantity and
the denominator equal to the
second quantity. In this case
first quantity is mangoes).
Ratio of mangoes to oranges =
Number of mangoes/Number of
oranges.
Step 3: Substitute the values of mangoes and
oranges.
Ratio of mangoes to oranges = 8/12.
Step 4: Simplify the expression.
(Hint: Divide both numerator and denominator by HCF
to obtain simplest form).
Ratio of mangoes to oranges
= 8/12 (divide both terms by HCF i.e. 4).**
= 2/3.
= 2 : 3.
Step 5: Write the answer.
Ratio of mangoes to oranges is 2 : 3.
(** HCF of 8 and 12 is 4)
Note: The answer can be written as: Ratio is 2 : 3 or
2/3 or 2 to 3