NCERT Exemplar Solutions Class 8 Mathematics Solutions for Rational Numbers - Exercise in Chapter 1 - Rational Numbers

Question 338 Rational Numbers - Exercise

Shalini has to cut out circles of diameter 1¼ cm from an aluminum strip of dimensions 8¾ cm by 1¼ cm. How many full circles can Shalini cut? Also calculate the wastage of the aluminum strip.


From the question it is given that,

Diameter of the circle = Breadth of the aluminium strip 1¼ cm = 5/4 cm

Length of aluminium strip = 8¾ cm = 35/4 cm

∴The number of full circles cut from the aluminum strip = (35/4) ÷ (5/4)

= (35/4) × (4/5)

= (7/1) × (1/1)

= 7 circles

Radius of circle = (5/ (4 × 2)) = 5/8 cm

Area to be cut by one circle = πr^2

= (22/7) × (5/8)^2

= (22/7) × (25/64) cm^2

Now, area to be cut by 7 full circles = 7 × (22/7) × (25/64)

= (22 × 25)/64

= 550/64 cm^2

Area of the aluminum strip = length × breadth

= (35/4) × (5/4) cm^2

= (175/16) cm^2

∴The wastage of aluminum strip = (175/16) – (550/64)

= (700 - 550)/64

= 150/64

= 75/32 cm^2

Video transcript
"hello students welcome to lido homework i'm kislee praveen your math tutor and today in this video i'm going to solve a math problem so let us see what the question is so there is a figure given here this is a diagram now what this diagram is about let us read shalini has to cut out circles of diameter one one by four centimeter from an aluminium strip of dimensions eight three by four centimeter by one three by four centimeter so using the first sentence we get the x uh understanding that this is a rectangle this rectangle is a aluminium foil and the length of the aluminum foil is eight three by four centimeter and width is one one by four centimeter now shalini has to cut circles from this aluminium foil okay and the diameter of that every circle needs to be one one by four centimeter so if you can observe this one by one one by four centimeter is also the diameter of the circle like this diameter and this diameter is nothing but also the width of the circle okay width of the rectangle is also diameter okay or let me just not write or it is width and diameter both okay so now what is the second sentence see it sees how many full circles can shalini cut also calculate the wastage of the aluminium strip so first we need to check how many circles can the shalini cut okay so obviously if you divide length by the width those many circles the shalini can cut right that is this width is one one by four so one width one one by four another one one by four another one one one by four across it lens will give you how many circles she can cut now why she can do that because diameter is not just the width but this part also it's a circle so even this is the diameter so from here to here that distance is the diameter of the circle okay again from here to here another diameter another diameter so uh if 8 3 by 4 is divided by 1 1 by 4 you get number of circles that can be number of diameters that can be cut across the length so let us do that okay so number sorry just a minute i need to erase this part and write number of circles is equal to 8 3 by 4 divided by 1 1 by 4. this further equals you need to convert it into improper fraction form so it was a 32 plus 3 it is 35 by 4 divided by 4 ones are 4 plus 1 5 by 4 now according to law of division you can change it into multiplication by finding the reciprocal of the second number so this becomes 4 by 5. now this 4 and 4 gets cancelled 5 ones are 5 5 7's are 35 so there are seven circles that can be cut okay so first part we have already answered now let us answer the second part also calculate the wastage of the aluminium strip so for that you need to find the area of this particular aluminium foil that is length into bread because it's rectangular minus total area of all the seven circles okay so whatever is left is the v stage okay so you can first find area of sec seven circles area of seven circles is equal to 7 into pi r square that is the area of 1 circle so this is 7 into pi is nothing but 22 by 7 into what is the radius of the circle its diameter by 2 so diameter is 5 by 4 that we clearly saw here so 5 by 4 by 2 so into 2 in the denominator you need to do that 2 times because it is r square by 4 into 2 so this 7 and this 7 will get cancelled to 11. so further it cannot be cancelled so numerator 11 into 5 into 5 is 275 and denominator is 32 so that much centimeter square is the area of all the seven circles put together okay so now let me erase some part the first question part and we can continue solving the second part of the question okay so coming back to that we need to find the area of the aluminium foil area of a l dot is aluminium foil is equal to length into bread so length is how much 5 by 4 that is the width and length is 35 by 4 so that is equal to 175 by 16 square centimeter so now you can easily calculate the wastage v stage is equal to area of i'm writing a rectangle minus area of 7 circles right so let me erase some more part of it and then continue solving so this becomes equal to area of rectangle is how much it is 175 by 16 minus area of 7 circles we got it as 275 by 32 so here the lcm of 16 and 32 is 32 so 16 2's are 32 so 175 into 2 will give you 350 minus 275 so the final answer is 75 by 32 square centimeter now you can convert it into decimal or mixed fraction form but this is the final answer i hope it's clear if you still have any doubt put a message in the comment section and subscribe to our channel for regular updates bye "
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