Math-for-all-Grades

**1. Express ratios always in simplest terms (lowest terms). **

The terms of a ratio are said to be in lowest terms, when they do not have any common factor other than 1.

Consider the ratio 150: 120

150: 120 = 15 × 10: 12 ×10 = 15: 12

Again, 15: 12 = 5 × 3: 4 × 3 = 5: 4

So, the H.C.F. (highest common factor) 30 (10×3) is removed from the two terms of the ratio 150: 120 to express the ratio in the simplest terms (lowest terms) 5: 4

Now, the H.C.F in the terms of the ratio 5: 4 is 1.

The ratio 5: 4 is said to be in lowest terms.

2. In a ratio, the order of the terms carries importance.

The ratio 1: 2 is not the same as 2:1

In the ratio 2: 1, we understand the first term is twice as greater as the second, whereas in the ratio 1: 2, the first is half as much as the second.

Example:

John is 50 years old and his son 25 years, today.

Then the ratio of the ages of the father and his son is 50: 25 = 2: 1

In words, we say “the father is twice as old as his son” today.

Whereas the ratio of the ages of the son and his father is 25: 50 = 1: 2, which

In words can be put as “the son is half as old as his father” today.

3. A ratio can be also expressed as a fraction.

The ratio 2: 3 can be also written as 2/3

The ratio 6: 8 can be also written as 6/8

Now, 6/8 is same as ¾ in lowest terms after removing the H.C.F. (highest common factor) 2 from the two terms 6 and 8

4. Two ratios are equal (or not equal) if their equivalent fractions are equal (or not equal)

The ratios 2: 3 and 12: 18 are equal as theirequivalent fractions are: 2/3 and 12/18. Now, 12/18 = (2 × 6)/ (3 × 6) = 2/3

5. A ratio does not change on multiplying (or dividing) the numerator and denominator by a same number:

The ratio ¾ does not change on multiplying the numerator and denominator by a same number:

¾ = (3 × 2)/ (4 × 2) = 6/8 or

¾ = (3 × 3)/ (4 × 3) = 9/12or

¾ = (3 × 5)/ (4 × 5) = 15/20 and so on.

The fractions 6/8, 9/12, 15/20 are all equal (as each of them reduces to ¾ on removing the respective HCF from the fractions)

6. When a same number is added to or deducted from the numerator and the denominator, then ratios will change

For example 2: 3 is not same as 2 + 2: 3 + 2, which is 4: 5.

Now, the fraction 2/3 is not equal to 4/5

Again, 2: 3 is not same as 2 – 1: 3 – 1, which is 1: 2

And, 2/3 is not equal to 1/2

**To compare ratios, write them as fractions and compare the fractions. **

Which of the two ratios is greater?

7: 8 or 12:14

Now, 7: 8 = 7/8 and 12: 14 = 6: 7 (on removing the HCF 2)

Is 7/8 > 6/7

Cross multiply to compare as:

Is 7 × 7 > 6 × 8,

i.e. is 49 > 48, Yes, it is .

Therefore, 7: 8 > 12: 14

Note:

Division is another method to compare fractions. Express ratios as fractions and perform division to compare the resulting quotients.

But cross multiplying is preferable to division of fractions.

**Ratios are simplified by removing the HCF from the terms of the ratio. **

The ratio 25: 35 is 5 × 5: 5 × 7 = 5: 7

**Similar quantities can be expressed as a ratio, only when they are in same units. **

The ratio of 3 feet to 3 yards is not 3: 3 = 1: 1

But, 3 feet = 90 cms and 3 yards = 100 cms.

Therefore, the ratio 3 feet: 3 yards is

90 cms: 100 cms = 9: 10

Finding LCM of denominators to express fractions as ratios:

**Express 2/3 , 3/4 and 4/5 as a ratio:**

The ratio is:

2/3: 3/4: 4/5

Now, find the LCM a 4 of the denominators: 3, 4 and 5.

LCM of 3, 4 and 5 = 3 × 4 × 5 = 60

Now, multiply each fraction with this LCM 60

(2/3) ×60: (3/4) × 60: (4/5)× 60 = 40: 45: 48

Suppose a father has $20.

He divides this money between his two sons in the parts $8 and $12

Then the ratio of the parts is 8: 12 = 2: 3.

Now, how do we find each of the two parts if the ratio of division and the money is given?

The ratio of the parts is 2: 3

Sum of the ratio terms is 2 + 3 = 5.

So, the first part is 2/5 th of the whole and

The second part is 3/5 th of the whole, i.e.

First is [2/5] × 20 = 8, and

The second is [3/5] × 20 = 12

**Let us generalize the above discussion into a formula as follows:**

Suppose a big number X is divided into two parts A and B in the ratio

p: q.

Then the first part **A = [p/ (p + q)] × X**

And the second part **B = [q/ (p + q)] × X**

**Example: **

**Divide 100 in the ratio 2: 3**

**Solution: **

The first part is [2/ (2 + 3)] × 100 = [2/5] × 100 = 40

The second part is [3/ (2 + 3)] × 100 = [3/5] × 100 = 60

**1. Example: A: B = 2: 3 and B: C = 4: 5, then Find A: B: C**

**Solution: **

To find A: B: C, the term B has to be same in the two ratios.

To make B same in the two ratios:

Multiply the first ratio terms with 4 (which B is in 4: 5)

Multiply the second ratio terms with 3 (which B is in 2: 3)

Now the first ratio A: B becomes 4 × (2: 3) = 8: 12, and

The second ratio B: C becomes 3 × (4: 5) = 12: 15, so

A: B = 8: 12 and B: C = 12: 15, therefore

A: B: C = 8: 12: 15

Short Cut:

**Suppose A: B = 2: 3 and A: C = 4: 5**

Then, consider B: A which is 3: 2

Now, B: A = 3: 2 and A: C = 4: 5

Let us make A same in the two ratios using the method discussed above.

Multiply B: A with 4 (which A is in 4:5)

So, B: A = 4 × (3: 2) = 12: 8, an d

Multiply A: C with 2 (which A is in 3: 2)

So, A: C = 2 × (4: 5) = 8: 10

Now, B: A: C = 12: 8: 10.

Rewrite this ratio for A: B: C as 8: 12: 10

**Suppose A: B is 2: 3 and B: C is 4: 5. **

Then A: C is found as below:

A: C as a fraction is A/C

But, A/C = (A/B) × (B/C)

So, A/C = (2/3) × (4/5) = 8/15

Therefore, A: C = 8: 15

**Suppose A: B = a: b, B: C = c: d and C: D = e: f**

**Short-Cut:**

**A: B: C: D = ace: bce: bde: bdf**

Suppose A: B = 2: 3 and B: C = 4: 5 and C: D = 6: 7.

Then, A: B: C: D = (2 × 4 × 6): (3 × 4 × 6): (3 × 5 × 6): (3 × 5 × 7)

= 48: 72: 90: 105.

On removing the HCF 3 from the terms, the ratio simplifies to

16: 24: 30: 35