Class 7 – Arithmetic

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Arithmetic
  • Ratio
  • Proportion
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Arithmetic
  • Percentages
  • Profit and loss
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Arithmetic
  • Simple interest
  • Average
  • Time and distance
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Ratio: The comparison of two quantities of the same kind in the same units is the fraction that one quantity is of the other is called ratio.

Thus, the ratio a is to b is the fraction written as a : b. In the ratio a : b, we call ‘a’ as the first term or antecedent and ‘ b’, as the second term or consequent.

Thus, in the ratio 7 : 4, we have
First term or antecedent = 7, Second term or consequent = 4.

Now let us understand its application through this example.

Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk.

So the ratio of flour to milk is 3 : 2

If you need to make pancakes for a LOT of people you might need 4 times the quantity, so you multiply the numbers by 4;

In other words, 12 cups of flour and 8 cups of milk.

The ratio is still the same, so the pancakes should be just as yummy.

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Facts about Ratio:

1. The ratio between two quantities of the same kind and in same units is obtained on dividing the first quantity by the second.

Examples: i) Ratio between 125 cm and 175 cm =
ii) Ratio between 18 years and 24 years

2. Ratio is a fraction. It has no units.

3. The quantities to be compared to form a ratio should be of the same kind. We cannot have a ratio between 16 years and ` 40.
Similarly, 25 cm and 200 g cannot form a ratio.

4. To find a ratio between two quantities of the same kind, both the quantities should be taken in the same units.

Examples:
i) Ratio between 75 g and 1 kg (1000 g) = 75 :
1000 =

ii) Ratio between 65 cm and 1 m
(since 1 m = 100 cm)
= 65 : 100 =

5. If each term of a ratio multiplied or divided by the same non-zero number, then the ratio remains the same.

Examples:

i) 3 : 4

ii)

Ratio in the simplest form (or in the lowest terms): A ratio a : b is said to be in simplest form, if H.C.F. of a and b is 1.

Examples:

i) The ratio 3 : 4 is in simplest form, since H.C.F. of 3 and 4 is 1.

ii) The ratio 12 : 16 is not in simplest form, since H.C.F. of 12 and 16 is 4.

Rule: To convert a ratio a : b in simplest form, divide a and b by the H.C.F of a and b.

Comparison of ratios: Since ratios are fractions, they can be compared similar to the way we compare fractions i.e. by converting them into equivalent like fractions, or by the cross product method.

Example: Which ratio is greater of 3 : 5 and 4 : 7?

Solution:

Let us take the cross product  ⇒  21 > 20

                                              ∴ .

Increase or decrease in a ratio: Suppose a quantity increases or decreases in the ratio a : b. Then, new quantity =  of the original quantity.

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The above pictures show the original and the enlarged picture frames.

Example: Express each of the following ratios in their simplest form:

i) 48 : 60             ii)

Solution:

i) 48 : 60; H.C.F of 48 and 60 = 12

ii)

 

Example: If A : B = 5 : 6 and B : C = 12 : 25, find A : C.

Solution: Since A : B = 5 : 6 and B : C = 12 : 25,
we have  and

Example: Two numbers are in the ratio 3 : 7 and their difference is 36. Find the numbers.

Solution: Let the smaller number be 3x and the greater number be 7x.

Then, 7x – 3x = 36  ⇒ 4x = 36 x = 9.

∴ Smaller number ;

   Greater number

Example: Divide ` 215 into three parts such that the first one is of the second and the ratio between second and third is 4 : 3.

Solution: Let the ratio of the three parts be a : b : c

Then,  a : b = 2 : 5.

Also, it is given that b : c = 4 : 3

Now a : b = 2 : 5 and b : c = 4 : 3

Let us make b, the same in each ratio, i.e. equal to 5 x 4 = 20

∴  a : b = 2 : 5 = 8 : 20 and b : c = 4 : 3 = 20 : 15

∴  a : b : c = 8 : 20 : 15

Sum of ratio terms = (8 + 20 + 15) = 43

∴ First part = `  = ` 40

Second part = `  = ` 100        Third part = `  = ` 75

Example: Which ratio is greater among 13 : 24 or 17 : 32?

Solution:

L.C.M. of 24 and 32 is

2

24, 32

2

12, 16

2

  6,  8

  3,  4

∴ LCM of (24, 32) is 96

∴ 13 : 24 = ; and 17 : 32 =

Clearly, . Hence, (13 : 24) > (17 : 32)

 

 

Types of ratios:

Compounded ratio: The compounded ratio of the two ratios a : b and c : d is ac : bd, and that of a : b, c :d and e : f is ace : bdf.

Duplicate ratio: The duplicate ratio of the ratio a : b is a2 : b2

Triplicate ratio: The triplicate ratio of the ratio a : b is a3 : b3.

ub duplicate ratio: The sub duplicate ratio of the ratio a : b is the ratio. So, the sub duplicate ratio of the ratio a2 : b2 is a : b.

Sub triplicate ratio: The sub triplicate ratio of the ratio a : b is . So, the sub triplicate ratio of the ratio a3 : b3 is a : b.

Reciprocal ratio: The reciprocal ratio of the ratio a : b is b : a .

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Part-to-Part” and “Part-to-Whole” Ratios: The examples so far have been “part-to-part” (comparing one part to another part). But a ratio can also show a part compared to the whole lot.  Example: There are 5 pups, 2 are boys, and 3 are girls

Part-to-Part: The ratio of boys to girls is 2 : 3 or
The ratio of girls to boys is 3 : 2 or

Part-to-Whole: The ratio of boys to all pups is 2 : 5 or
The ratio of girls to all pups is 3 : 5 or .

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