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Fractions

Class 9th Mathematics Part 1 Kerala Board Solution
Questions Pg-24
  1. 1^2 + 1/1+1 = 1 2^2 + 2/2+2 = 1 1/2 3^2 + 3/3+3 = 2 4^2 + 4/4+4 = 2 1/2 Explain each of…
  2. 2^2 - 2/2-1 = 2 3^2 - 3/3-1 = 3 4^2 - 4/4-1 = 4 5^2 - 5/5-1 = 5 Explain each of the…
  3. 2^2 - 1/2-1 = 3 3^2 - 1/3-1 = 4 4^2 - 1/4-1 = 5 5^2 - 1/5-1 = 6 Explain each of the…
Questions Pg-28
  1. Look at this method of making a pair of equal fraction from another such pair. 1/3 =…
  2. Look at these calculations: 1/2 = 2/4 (3 x 1) + (4 x 2)/(3 x 2) + (4 x 4) = 11/22 = 1/2…
  3. The sum of the square of a number and one, divided by the difference of 1 from the…
  4. The sum of a number and its square is one and a half times their difference. What is…
Questions Pg-34
  1. Find the larger of each pair of fractions below, without multiplication: 13/17 , 14/15…
  2. Find the larger of each pair of fractions below, without pen and paper. 3/5 , 8/13 3/5…
  3. i) Find three fractions larger than 1/3 and smaller than 1/2 ii) Find three such…
  4. From a fraction, a new fraction is formed by adding the same natural number to both the…
Questions Pg-38
  1. 1 - 1/3 = 2/3 = 2/2^2 - 1 1/2 - 1/4 = 2/8 = 2/3^2 - 1 1/3 - 1/5 = 2/15 = 2/4^2 - 1 Find…
  2. 1/2 + 2/1 = 5/2 = 2 + 1/1 x 2 2/3 + 3/2 = 13/6 = 2 + 1/2 x 3 3/4 + 4/3 = 25/12 = 2 +…
  3. ll 1/2 + 1/3 = 5/6 & 3/2 - 2/3 = 5/6 1/3 + 1/4 = 7/12 & 4/3 - 3/4 = 7/12 1/4 + 1/5 =…
  4. ll 4 1/4 - 1 1/2 = 3 & 4 1/2 - 1 1/2 = 3 5 1/3 - 1 1/3 = 4 & 5 1/3 - 1 1/3 = 4 6 1/4 -…
Questions Pg-45
  1. Find the fraction of denominator is a power of 10 equal to each of the fractions below,…
  2. 5/3 Find fractions of denominators which are power of 10, getting closer and closer to…
  3. 3/11 Find fractions of denominators which are power of 10, getting closer and closer…
  4. 23/11 Find fractions of denominators which are power of 10, getting closer and closer…
  5. 1/13 Find fractions of denominators which are power of 10, getting closer and closer…
  6. i) Explain using algebra, that the fractions 1/10 , 11/100 , 111/1000 , gets closer and…
Questions Pg-24
Question 1.

Explain each of the patterns below and write the general principle in algebra.
Answer:





Generally, we can write,


where, n can be any natural number.
Question 2.

Explain each of the patterns below and write the general principle in algebra.
Answer:





Generally, we can write,


where, n can be any natural number.
Question 3.

Explain each of the patterns below and write the general principle in algebra.
Answer:

using identity, a2 - b2 = (a + b)(a-b), in the numerator.






Generally, we can write,


where, n can be any natural number.
Questions Pg-28
Question 1.

Look at this method of making a pair of equal fraction from another such pair.



i) Check some more pairs of equal fractions. By interchanging the numerator of one with the denominator of the other, do you get equal fractions?

ii) Write this as a general principle using algebra and explain it.
Answer:

i)




So, in all these examples we get equal fractions.


ii) General principle using algebra,


We know that,


When then, aq = pb ………. (1)


Now, on dividing both sides by pq, we get
Question 2.

Look at these calculations:



i) Take some more fractions equal to and form fractions by multiplying the numerators and denominators by 3 and 4 and adding.

Do you get fractions equal to

ii) Take some other pairs of equal fractions and check this

iii) In all these, instead of multiplying numerators and denominators by 3 and 4, multiply by some other numbers and add. Do you still get equal fractions?

iv) Explain why, if the fraction is equal to the fraction then for any pair of natural numbers m and n, the fractions is equal to


Answer:

i)



We observe that, in these cases also, we obtain


ii)




We observe in these cases also, we obtain equal fraction,


iii) let us now take 2 and 5 instead of 3 and 4 in part(ii).





So, again we get equal fractions.


iv) We know that,


When then, aq = pb ………. (1)


∴ for


(ma + np) × b = a × (mb + nq) →this must be satisfied


∴ (ma + np) × b = mab + npb ………… (2)


And a × (mb + nq) = mab + nqa


= mab + npb ………… using (1)


∴ a × (mb + nq) = mab + npb ………….. (3)


Since, (2) is equal to (3)


∴



Question 3.

The sum of the square of a number and one, divided by the difference of 1 from the square gives What is the number?
Answer:

Let the number be x.


⇒


⇒ 220(x2 + 1) = 221(x2 – 1)


⇒ 220x2 + 220 = 221x2 – 221


⇒ x2 = 441


⇒ x = ±21


∴ The required number can be 21 or -21.
Question 4.

The sum of a number and its square is one and a half times their difference. What is the number?
Answer:

Let the number be x.


As per the condition,


⇒


⇒ 2x + 2x2 = x – x2


⇒ 3x2 + x = 0


⇒


⇒ x = 0 or



Questions Pg-34
Question 1.

Find the larger of each pair of fractions below, without multiplication:
Answer:

We know that,


When the denominators of the fractions are equal ,


then the fraction having greater numerator is larger than


the other. Example:


Also, When the numerators of the fractions are equal ,


then the fraction having smaller denominator is larger


than the other. Example:


on combining these two statements we can conclude that,


The fraction having larger numerator and also smaller denominator is greater.


Now,


i) Among and


since, 14 is greater than 13 and 15 is smaller than 17.


∴


ii) Among and


since, 13 is greater than 11 and 17 is smaller than 18.


∴


iii) Among and


since, 14 is greater than 11 and 15 is smaller than 18.


∴



Question 2.

Find the larger of each pair of fractions below, without pen and paper.
Answer:

i)


We know that,


when aq<pb


3 × 13 = 39


5 × 8 = 40


Since, 39<40


∴


ii)


We know that,


when aq>pb


3 × 11 = 33


5 × 6 = 30


Since, 33>30


∴


iii)


In adding 3 to both numerator and denominator gives and since, 98<99


∴ ……… using, , when b>a
Question 3.

i) Find three fractions larger than and smaller than

ii) Find three such fractions, all with the denominator 24.

iii) Find three such fractions, all with the numerator 4.
Answer:

i) we know that, when then …….. (1)



….…….using (1)


⇒ …………using (1)


∴ are the required fractions.


ii) Lets make the denominator of and equal to 24.




Now, we can say that,



∴ are the required fractions.


iii) Lets make the numerator of and equal to 4.




Now, we can say that,



∴ are the required fractions.
Question 4.

From a fraction, a new fraction is formed by adding the same natural number to both the numerator and the denominator.

i) In what kind of fractions does this give a larger fraction?

ii) In what kind of fractions does this give a smaller fraction?
Answer:

Let the fraction be


Now, fraction obtained by adding a natural number n to both numerator and denominator =


i) when b>a


Proof:


⇒ a(b + n) < (a + n)b


⇒ ab + an < ab + bn


⇒ an < bn


⇒ a < b


ii) when b<a


Proof:


⇒ a(b + n) > (a + n)b


⇒ ab + an > ab + bn


⇒ an > bn


⇒ a > b
Questions Pg-38
Question 1.

Find the general principle of each of the patterns below and explain it using algebra.
Answer:

On observing the pattern we get general formula as,



To prove this algebraically,


We know that so,




Using, (a-b)(a + b) = a2 – b2



Hence, proved!
Question 2.

Find the general principle of each of the patterns below and explain it using algebra.
Answer:

On observing the pattern we get general formula as



To prove this algebraically,


We know that so,








Hence, proved!
Question 3.

Find the general principle of each of the patterns below and explain it using algebra.
Answer:

On observing the pattern we get general formula as



To prove this algebraically,


We know that so,


……..(1)




∵ using, a2-b2 = (a-b)(a + b)



…………….(2)


From (1) and (2) , we get



Hence, proved!
Question 4.

Find the general principle of each of the patterns below and explain it using algebra.
Answer:

On observing the pattern we get general formula as



To prove this algebraically,


We know that so,






………………….(1)





…………………….(2)


From (1) and (2) , we get



Hence, proved!
Questions Pg-45
Question 1.

Find the fraction of denominator is a power of 10 equal to each of the fractions below, and then write their decimal forms:
Answer:

i)


ii)


iii)


iv)



Question 2.

Find fractions of denominators which are power of 10, getting closer and closer to each of the fractions below and then write their decimal form.
Answer:



⇒


Thus, we get a fraction of denominator 10, close to


Now, to get a fraction of denominator 100, close to ,


start with




⇒


Similarly, we can have




⇒


We observe that, the fractions get closer and closer to


∴ we can write,
Question 3.

Find fractions of denominators which are power of 10, getting closer and closer to each of the fractions below and then write their decimal form.
Answer:



⇒


Thus, we get a fraction of denominator 10, close to


Now, to get a fraction of denominator 100, close to ,


start with




⇒


Similarly, we can have




⇒


We observe that, the fractions get closer and closer to


∴ we can write,
Question 4.

Find fractions of denominators which are power of 10, getting closer and closer to each of the fractions below and then write their decimal form.
Answer:



⇒


Thus, we get a fraction of denominator 10, close to


Now, to get a fraction of denominator 100, close to ,


start with




⇒


Similarly, we can have




⇒




⇒


We observe that, the fractions get closer and closer to


∴ we can write,
Question 5.

Find fractions of denominators which are power of 10, getting closer and closer to each of the fractions below and then write their decimal form.
Answer:



⇒


Thus, we get a fraction of denominator 10, close to


Now, to get a fraction of denominator 100, close to ,


start with




⇒


Similarly, we can have




⇒


We observe that, the fractions get closer and closer to


∴ we can write,
Question 6.

i) Explain using algebra, that the fractions gets closer and closer to

ii) Using the general principle above on single digit numbers, find the decimal forms of (why are left out in this?)

iii) What can we say in general about those decimal forms in which a single digit repeats?
Answer:

i) We can easily see that





Since,


Thus, get closer and closer to


ii) From part (i),


We can write


∴






are excluded because they in these fractions numerator and denominator have common factors.


iii) The decimal forms in which a single digit repeat are generally those in which we get the same remainder after each step. These forms are called repeating or recurring decimal forms.