Ratios

A ratio is a comparison of quantities with the same units, typically represented by two or more numbers separated by a colon (e.g., 3:5:7).

Ratios can be used to describe relationships in everyday life and in various mixtures. For example, if you have two red sweets and four blue sweets, the ratio of red sweets to blue sweets can be expressed as 2:4.

In a mixture of cement, sand, and gravel, if there are 3 parts cement, 5 parts sand, and 7 parts gravel, the ratio representing this mixture would be 3:5:7.

Simplifying Ratios

Ratios can be simplified by finding common factors and reducing them to their lowest form. For example, the ratio 12:18 can be simplified to 2:3 by dividing both numbers by their common factor, 6. This process is similar to simplifying fractions.

Let’s look at an example:

Simplify the ratio 16:24.

Both numbers have a common factor of 8. Dividing each number by 8, we get:

\frac{16}{8} : \frac{24}{8} = 2:3

So, the simplified ratio is 2:3.

Converting Ratios

Ratios can be converted to fractions and percentages to better understand the relative quantities in a given situation.

To convert a ratio to fractions, first sum the numbers in the ratio. If the ratio is a:b:c, calculate a+b+c. The fractions for each component would be \frac{a}{a+b+c}, \frac{b}{a+b+c}, and \frac{c}{a+b+c}.

To convert a fraction to a percentage, multiply the fraction by 100.

Let’s look at an example:

Convert the ratio 3:5:7 to fractions and percentages.

First, add the numbers in the ratio: 3+5+7=15. Then, calculate the fractions for each component: \frac{3}{15}, \frac{5}{15}, and \frac{7}{15}.

Next, convert each fraction to a percentage:

  • \frac{3}{15}\times 100 = 20%
  • \frac{5}{15}\times 100 = 33.33%
  • \frac{7}{15}\times 100 = 46.67%

Equivalent Ratios

Equivalent ratios are ratios that represent the same relationship between two quantities, even though the actual numbers in the ratios may be different. If two ratios have the same value, then they are equivalent.

For example, the ratios 2:3 and 4:6 are equivalent because the relationship between the numbers in each ratio is the same (multiplying the first number by 2 gives the second number).

How to Find Equivalent Ratios

To find equivalent ratios, you can either multiply or divide both terms of a given ratio by the same non-zero number. This will maintain the proportion and create a new ratio with the same relationship between the quantities.

Let’s look at an example:

Find three equivalent ratios for the ratio 6:10.

We can multiply or divide both terms of the ratio by different non-zero numbers to find equivalent ratios.

  • Multiply by 2: (6 \times 2):(10 \times 2) = 12:20
  • Multiply by 3: (6 \times 3):(10 \times 3) = 18:30
  • Divide by 2: (6 \div 2):(10 \div 2) = 3:5

So, three equivalent ratios for the ratio 6:10 are 12:20, 18:30, and 3:5.

Examples

Example 1:

Sam and Ram share £200 in the ratio 3 : 2. How much does Sam get?

To find Sam’s share, we first determine the total number of parts in the ratio and then find the value of each part. Finally, we calculate Sam’s share based on the number of parts he gets.

Total parts in the ratio: 3 + 2 = 53 + 2 = 5 Each part’s value: \frac{200}{5} =\frac{200}{5} =£4040

Sam’s share (3 parts): 3 \times £40 =3 \times £40 = £120120

Sam gets £120.

Example 2:

Adam has £50 to share among his three children: Bill, Claire, and Diana, in the ratio 4:5:1. How much will each child receive?

First, add the numbers in the ratio: 4+5+1=104+5+1=10. Then, calculate the share for each child:

  • Bill: \dfrac{4}{10}\times\dfrac{4}{10}\times £50=50=£2020
  • Claire: \dfrac{5}{10}\times\dfrac{5}{10}\times £50=50=£2525
  • Diana: \dfrac{1}{10}\times\dfrac{1}{10}\times £50=50=£55

Example 3:

Alan and Bernard share £230 in the ratio 1\frac{1}{2} : 2\frac{1}{3}. How much does Bernard get?

First, we need to simplify the ratio 1\frac{1}{2} : 2\frac{1}{3}1\frac{1}{2} : 2\frac{1}{3}. Convert it into improper fractions: this is \frac{3}{2} : \frac{7}{3}\frac{3}{2} : \frac{7}{3}.

Now, we need to find a common denominator for the two fractions. The least common multiple of 2 and 3 is 6. So, we multiply both improper fractions by 6 to eliminate the denominators:

\frac{3}{2} : \frac{7}{3} =\frac{3}{2} \times 6 : \frac{7}{3} \times 6 = 9 : 14\frac{3}{2} : \frac{7}{3} =\frac{3}{2} \times 6 : \frac{7}{3} \times 6 = 9 : 14

The simplified ratio is 9 : 149 : 14. Now, we can find the share of Bernard:

Total parts in the ratio: 9 + 14 = 239 + 14 = 23 Each part’s value: \frac{230}{23} =\frac{230}{23} = £1010

Bernard’s share (14 parts): 14 \times £10 =14 \times £10 = £140140

Bernard gets £140.

Example 4:

Alan and Bernard share a certain amount of money in the ratio 2 : 3. Given that Bernard receives £15 more than Alan. How much does Alan receive? And how much does Bernard receive?

Let’s denote the total amount of money shared as T, Alan’s share as A, and Bernard’s share as B. We have the following relationships:

1. \frac{A}{T} = \frac{2}{5}\frac{A}{T} = \frac{2}{5} and \frac{B}{T} = \frac{3}{5}\frac{B}{T} = \frac{3}{5}

2. B = A +B = A +£1515

First, we need to find the total amount of money (T) by using the information that Bernard receives £15 more than Alan. Since the difference between the two shares is \frac{1}{5}\frac{1}{5} of the total amount, we can write:

\frac{1}{5} \times T =\frac{1}{5} \times T = £1515

Now we can solve for T:

T = 15 \times 5 =T = 15 \times 5 = £7575

Now that we know the total amount of money shared, we can find Alan’s and Bernard’s shares:

Alan: A = \frac{2}{5} \timesA = \frac{2}{5} \times £75 = £3075 = £30 Bernard: B = \frac{3}{5} \times £75 =B = \frac{3}{5} \times £75 = £4545

So, Alan receives £30, and Bernard receives £45.

Example 5:

Given that Ahmad’s age : Brown’s age = 3 : 4 and Brown’s age : Caulas’ age = 4 : 7. Find the ratio of Ahmad’s age : Brown’s age : Caulus’ age.

Denoting Ahmad, Brown, and Caulus by A, B, and C respectively, we have: A : B = 3 : 4A : B = 3 : 4 and B : C = 4 : 7B : C = 4 : 7.

To find the combined ratio, we need to ensure that the value for Brown’s age in both ratios is equal. To do this, we can cross-multiply the ratios:

(A : B) \times (B : C) = (3 : 4) \times (4 : 7)(A : B) \times (B : C) = (3 : 4) \times (4 : 7)

Now, multiply the respective terms: A : B : C = (3 \times 4) : (4 × 4) : (4 \times 7)A : B : C = (3 \times 4) : (4 × 4) : (4 \times 7)

= 12 : 16 : 28= 12 : 16 : 28

We can simplify this ratio further by dividing each term by the greatest common divisor, which is 4:

A : B : C = \frac{12}{4} : \frac{16}{4} : \frac{28}{4} = 3 : 4 : 7A : B : C = \frac{12}{4} : \frac{16}{4} : \frac{28}{4} = 3 : 4 : 7

The ratio of Ahmad’s age : Brown’s age : Caulus’ age is 3 : 4 : 73 : 4 : 7.

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