Comparing Quantities

8.1 Introduction

· In our daily life, there are many occasions when we compare two quantities. They may be heights, weights, salaries, marks etc.

· Suppose we are comparing heights of Heena and Amir.

We find that

1. Heena is two times taller than Amir. Or

2. Amir’s height is of Heena’s height

· Eg: While comparing heights of two persons with heights 150 cm and 75 cm,

We write it as the ratio 150: 75 or 2: 1.

· Yet another example is where we compare speeds of a Cheetah and a Man.

The speed of a Cheetah is 6 times the speed of a Man.

Or

The speed of a Man is of the speed of the Cheetah.

● Thus, we see that the ratio for two different comparisons may be the same.

● Remember that to compare two quantities, the units must be the same.

● A ratio has no units.

Example 1 Find the ratio of 3 km to 300 m.

Solution

First convert both the distances to the same unit.

So,

3 km = 3 × 1000 m = 3000 m.

Thus, the required ratio, 3 km: 300 m is 3000: 300 = 10: 1.

8.2 Equivalent Ratios

· Different ratios can also be compared with each other to know whether they are equivalent or not.

· To do this, we need to write the ratios in the form of fractions and then compare them by converting them to like fractions.

· If these like fractions are equal, we say the given ratios are equivalent.

Example 2 Are the ratios 1:2 and 2:3 equivalent?

Solution

To Check This, We Need To Know Whether.

We Have,

;

We Find That 3 /6 < 4 /6,

Which Means That 1 /2 < 2/ 3.

Therefore, The Ratio 1:2 Is Not Equivalent To The Ratio 2:3.

Example 3 Following Is The Performance Of A Cricket Team In The Matches It Played:

In Which Year Was The Record Better? How Can You Say So?

Solution

Last Year, Wins: Losses = 8: 2 = 4: 1

This Year, Wins: Losses = 4: 2 = 2: 1

Obviously, 4: 1 > 2: 1 (In Fractional Form, 4/ 1> 2/ 1)

Hence, We Can Say That The Team Performed Better Last Year.

In Class VI, We Have Also Seen The Importance Of Equivalent Ratios. The Ratios Which Are Equivalent Are Said To Be In Proportion. Let Us Recall the Use of Proportions

Keeping Things in Proportion And Getting Solutions

Example 4

A Map Is Given With A Scale Of 2 Cm = 1000 Km. What Is The Actual Distance Between The Two Places In Kms, If The Distance In The Map Is 2.5 Cm?

Solution

· Arun Does It Like This :

Let Distance = X Km

Then, 1000: X = 2: 2.5

X=2/2.5

· Meera Does It Like This:

2 Cm Means 1000 Km.

So, 1 Cm Means 1000/2 Km

Hence, 2.5 Cm Means 1000/2× 2.5 Km = 1250

Arun Has Solved It By Equating Ratios To Make Proportions And Then By Solving The Equation.

Meera Has First Found The Distance That Corresponds To 1 Cm And Then Used That To Find What 2.5 Cm Would Correspond To. She Used The Unitary Method.

Let Us Solve Some More Examples Using The Unitary Method.

Example 5 6 Bowls Cost 90. What Would Be The Cost Of 10 Such Bowls?

Solution

Cost Of 6 Bowls Is 90.

Therefore, Cost Of 1 Bowl = 90/ 6

Hence, Cost Of 10 Bowls = 90 /6 × 10 = 150

Example 6 The Car That I Own Can Go 150 Km With 25 Liters Of Petrol. How Far Can It Go With 30 Liters Of Petrol?

Solution

With 25 Liters Of Petrol, The Car Goes 150 Km.

With 1 Liter The Car Will Go 150 /25 Km.

Hence, With 30 Liters of Petrol It Would Go ×30km = 180 Km

· In This Method, We First Found The Value For One Unit Or The Unit Rate. This Is Done By The Comparison Of Two Different Properties.

· For Example, When You Compare Total Cost To Number Of Items, We Get Cost Per Item Or If You Take Distance Travelled To Time Taken, We Get Distance Per Unit Time.

· Thus, You Can See That We Often Use ‘Per’ To Mean For Each. For Example, Km Per Hour, Children Per Teacher Etc., Denote Unit Rates

8.3 Percentage – Another Way of Comparing Quantities

Percentages Are Numerators of Fractions With Denominator 100 And Have Been Used In Comparing Results

8.3.1 Meaning of Percentage

● Per Cent Is Derived From Latin Word ‘Per Centum’ Meaning ‘Per Hundred’

● Per Cent Is Represented By The Symbol % And Means Hundredths Too.

● That Is 1% Means 1 Out Of Hundred Or One Hundredth.

It Can Be Written As: 1% = 1/ 100 = 0.01

To Understand This, Let Us Consider The Following Example.

Rina Made A Table Top Of 100 Different Coloured Tiles. She Counted Yellow, Green, Red and Blue Tiles Separately and Filled the Table Below. Can You Help Her Complete The Table?

Percentages When Total Is Not Hundred

· In All These Examples, The Total Number Of Items Add Up To 100.

· For Example, Rina Had 100 Tiles In All, There Were 100 Children And 100 Shoe Pairs. How Do We Calculate Percentage Of An Item If The Total Number Of Items Do Not Add Up To 100? In Such Cases, We Need To Convert The Fraction To An Equivalent Fraction With Denominator 100. Consider The Following Example.

1) You Have A Necklace With Twenty Beads In Two Colours.

· We See That These Three Methods Can Be Used To Find The Percentage When The Total Does Not Add To Give 100.

· In The Method Shown In The Table, We Multiply The Fraction By 100 100. This Does Not Change The Value Of The Fraction. Subsequently, Only 100 Remains in the Denominator. Anwar Has Used The Unitary Method.

· Asha Has Multiplied By 5 /5 To Get 100 In The Denominator.

· You Can Use Whichever Method You Find Suitable. May Be, You Can Make Your Own Method Too.

8.3.2 Converting Fractional Numbers to Percentage

· Fractional Numbers Can Have Different Denominator. To Compare Fractional Numbers, We Need A Common Denominator And We Have Seen That It Is More Convenient To Compare If Our Denominator Is 100. That Is, We Are Converting The Fractions To Percentages.

· Let Us Try Converting Different Fractional Numbers To Percentages.

Example 7 Write 1/3 As Per Cent.

Solution

We Have,

Example 8 Out Of 25 Children In A Class, 15 Are Girls. What Is The Percentage Of Girls?

Solution

Out Of 25 Children, There Are 15 Girls.

Therefore, Percentage of Girls = 15/25 × 100 = 60.

There Are 60% Girls In The Class.

Example 9 Convert 5/4 To Per Cent.

Solution

We Have, 5/ 4 = 5/ 4 100%

= 125 %

8.3.3 Converting Decimals to Percentage

We Have Seen How Fractions Can Be Converted To Per Cents. Let Us Now Find How Decimals Can Be Converted To Per Cents.

Example 10 Convert The Given Decimals To Per Cents:

(A) 0.75 (B) 0.09 (C) 0.2

Solution

(A) 0.75 = 0.75 × 100 %

= 75 /100 × 100 %

= 75%

(B) 0.09 = 9/100 = 9 %

8.3.4 Converting Percentages to Fractions or Decimals

8.3.5 Fun With Estimation Percentages Help Us To Estimate The Parts Of An Area.

Example 11 What Per Cent Of The Adjoining Figure Is Shaded?

Solution

We First Find The Fraction Of The Figure That Is Shaded. From This Fraction, The Percentage Of The Shaded Part Can Be Found. You Will Find That Half Of The Figure Is Shaded.

And,

Thus, 50 % Of The Figure Is Shaded.

8.4 Use of Percentages

8.4.2 Converting Percentages to “How Many”

Example 12 A Survey Of 40 Children Showed That 25% Liked Playing Football. How Many Children Liked Playing Football?

Solution

Here, The Total Number Of Children Are 40.

Out of these, 25% like Playing Football.

Meena And Arun Used The Following Methods To Find The Number.

You Can Choose Either Method

· Arun Does It Like This

Out of 100, 25 Like Playing Football

So Out Of 40, Number of Children Who Like Playing Football

= 25/100 × 40 = 10

· Meena Does It Like This

25% of 40 = 25 × 40 100 = 10

Example 13 Rahul Bought A Sweater And Saved 200 When A Discount Of 25% Was Given. What Was The Price Of The Sweater Before The Discount?

Solution

Rahul Has Saved 200 When Price Of Sweater Is Reduced By 25%. This Means That 25% Reduction In Price Is The Amount Saved By Rahul. Let Us See How Mohan And Abdul Have Found The Original Cost Of The Sweater.

· Mohan’s Solution:

25% of the Original Price = 200

Let the Price (In) Be P

So, 25% of P = 200 Or 25/100 P= 200

Or, P/4 =200 or P = 200 × 4

Therefore, P = 800

· Abdul’s Solution:

25 Is Saved For Every 100

Amount for Which 200 Is Saved = 100/25 × 200 = 800

Thus Both Obtained The Original Price Of Sweater As 800.

8.4.3 Ratios To Percent’s

Sometimes, Parts Are Given To Us In The Form Of Ratios And We Need To Convert Those To Percentages.

Example 14

Reena’s Mother Said, To Make Idlis, You Must Take Two Parts Rice and One Part Urad Dal. What Percentage Of Such A Mixture Would Be Rice And What Percentage Would Be Urad Dal?

Solution

In Terms Of Ratio We Would Write This as Rice: Urad Dal = 2: 1.

Now, 2 + 1=3 Is The Total Of All Parts. This Means 2/ 3 Part Is Rice and 1/ 3 Part Is Urad Dal.

Then, Percentage of Rice Would Be

Percentage of Urad Dal Would Be

Example 15

If 250 Is To Be Divided Amongst Ravi, Raju And Roy, So That Ravi Gets Two Parts, Raju Three Parts And Roy Five Parts. How Much Money Will Each Get? What Will It Be In Percentages?

Solution

The Parts Which The Three Boys Are Getting Can Be Written In Terms Of Ratios As 2: 3:5. Total Of The Parts Is 2 + 3 + 5 = 10.

8.4.4 Increase or Decrease as Per Cent

If The Population Of A State Increased From 5, 50, 000 To 6, 05, 000. Then The Increase In Population Can Be Understood Better If We Say, The Population Increased By 10 %. How Do We Convert The Increase Or Decrease In A Quantity As A Percentage Of The Initial Amount? Consider The Following Example.

Example 16 A School Team Won 6 Games This Year Against 4 Games Won Last Year. What Is The Per Cent Increase?

Solution

The Increase in the Number of Wins (Or Amount of Change) = 6 – 4 = 2.

Percentage Increase = = Amount of Change / Original Amount or Base

=Increase in the Number of Wins /Original Number of Wins

=

= 50

Example 17

The Number Of Illiterate Persons In A Country Decreased From 150 Lakhs To 100 Lakhs In 10 Years. What Is The Percentage Of Decrease?

Solution

Original Amount = The Number of Illiterate Persons Initially = 150 Lakhs.

Amount of Change = Decrease In the Number of Illiterate Persons

= 150 – 100 = 50 Lakhs Therefore,

The Percentage of Decrease = Amount of Change /Original Amount

=

8.5 Prices Related To an Item or Buying and Selling

● The Buying Price Of Any Item Is Known As Its Cost Price. It Is Written In Short As CP.

● The Price At Which You Sell Is Known As The Selling Price Or In Short SP.

● You Can Decide Whether The Sale Was Profitable Or Not Depending On The CP And SP.

● If CP < SP Then You Made A Profit = SP – CP.

● If CP = SP Then You Are In A No Profit No Loss Situation.

● If CP > SP Then You Have A Loss = CP – SP.

· Let Us Try To Interpret The Statements Related To Prices Of Items.

1. A Toy Bought For 72 Is Sold At 80.

Let Us Consider The First Statement.

The Buying Price (Or CP) Is 72 And The Selling Price (Or SP) Is 80. This Means SP Is More Than CP.

Hence Profit Made = SP – CP = 80 – 72 = 8

8.5.1 Profit or Loss as A Percentage

● The Profit Or Loss Can Be Converted To A Percentage. It Is Always Calculated On The CP.

For The Above Examples, We Can Find the Profit % or Loss %.

Let Us Consider The Example Related To The Toy.

We Have CP = 72, SP = 80, Profit = 8. To Find The Percentage Of Profit, Neha And Shekhar Have Used The Following Methods.

· Neha Does It This Way:

Profit Per Cent = Profit/CP

· Shekhar Does It This Way:

On 72 The Profit is 8

On 100, Profit = 8/ 72 ×100

Thus, Profit Per Cent = 11 1/ 9

Thus, The Profit is 8 And Profit Per Cent Is 11 1/9.

· Similarly You Can Find The Loss Per Cent In The Second Situation.

Here, CP = 120, SP = 100.

Therefore, Loss = 120 – 100 = 20

Now We See That Given Any Two Out Of The Three Quantities Related To Prices That Is, CP, SP, Amount Of Profit Or Loss Or Their Percentage, We Can Find The Rest.

Example 18 The Cost of a Flower Vase is 120. If The Shopkeeper Sells It At A Loss Of 10%, Find The Price At Which It Is Sold.

Solution

We Are Given That CP = 120 and Loss Per Cent = 10. We Have To Find the SP

· Sohan Does It Like This:

Loss Of 10% Means If CP is 100,

Loss is 10

Therefore, SP Would Be (100 – 10) = 90

When CP is 100, SP is 90

Therefore, If CP Were 120 Then

· Anandi Does It Like This :

Loss Is 10% of the Cost Price

= 10% of 120

Therefore

SP = CP – Loss

= 120 – 12 = 108

Thus, By Both Methods We Get The SP as 108.

Example 19 Selling Price of a Toy Car is 540. If The Profit Made By Shopkeeper Is 20%, What Is The Cost Price Of This Toy?

Solution

We Are Given That SP = 540 and the Profit = 20%. We Need To Find The CP.

· Amina Does It Like This;

20% Profit Will Mean If CP is 100, Profit is 20

Therefore, SP = 100 + 20 = 120

Now, When SP is 120, then CP is 100.

Therefore, When SP Is 540,

Then CP = 450

· Arun Does It Like This;

Profit = 20% of CP and SP = CP + Profit

So, 540 = CP + 20% of CP

= CP + 20 /100 × CP = CP = 6/5 CP.

Therefore, or 450 = CP

Thus, By Both Methods, The Cost Price is 450.

8.6 Charge Given On Borrowed Money or Simple Interest

● The Money You Borrow Is Known As Sum Borrowed Or Principal.

● This Money Would Be Used By The Borrower For Some Time Before It Is Returned. For Keeping This Money For Some Time The Borrower Has To Pay Some Extra Money To The Bank. This Is Known As Interest.

● You Can Find The Amount You Have To Pay At The End Of The Year By Adding The Sum Borrowed And The Interest. That Is,

Amount = Principal + Interest.

● Interest Is Generally Given In Per Cent For A Period Of One Year. It Is Written As Say 10% Per Year Or Per Annum Or In Short As 10% P.A. (Per Annum).

● 10% P.A. Means on Every 100 Borrowed, 10 Is the Interest You Have To Pay For One Year. Let Us Take An Example And See How This Works.

Example 20 Anita Takes A Loan Of 5,000 At 15% Per Year As Rate Of Interest. Find The Interest She Has To Pay At The End Of One Year.

Solution

The Sum Borrowed = 5,000,

Rate of Interest = 15% per Year.

This Means If 100 Is Borrowed, She Has To Pay 15 As Interest For One Year.

If She Has Borrowed 5,000, Then the Interest She Has To Pay For One Year =

=

So, At the End of the Year She Has To Give An

Amount of 5,000 + 750 = 5,750.

We Can Write A General Relation To Find Interest For One Year.

Take P as the Principal or Sum And R % As Rate Per Cent Per Annum.

Now On Every 100 Borrowed, the Interest Paid Is R

Therefore, On P Borrowed, the Interest Paid For One Year Would Be

RP/100=PR/100

8.6.1 Interest for Multiple Years

● If The Amount Is Borrowed For More Than One Year The Interest Is Calculated For The Period The Money Is Kept For.

● For Example, If Anita Returns The Money At The End Of Two Years And The Rate Of Interest Is The Same Then She Would Have To Pay Twice The Interest

I.E., 750 for the First Year and 750 for the Second.

● This Way Of Calculating Interest Where Principal Is Not Changed Is Known As Simple Interest. As The Number Of Years Increase The Interest Also Increases.

● For 100 Borrowed For 3 Years At 18%, The Interest To Be Paid At The End Of 3 Years Is 18 + 18 + 18 = 3 × 18 = 54.

● We Can Find The General Form For Simple Interest For More Than One Year.

We Know That On a Principal of P at R% Rate of Interest per Year, the Interest Paid For One Year Is RP/100

● Therefore, Interest I Paid For T Years Would Be

TRP/100 = PRT/100 or PTR/100

Or

And Amount You Have To Pay At the End of T Years Is A = P + I

● Just As In The Case Of Prices Related To Items, If You Are Given Any Two Of The Three Quantities In The Relation

I= PT R /100, You Could Find The Remaining Quantity.

Example 21 If Manohar Pays An Interest of 750 For 2 Years On A Sum of 4,500, Find The Rate Of Interest.

Solution 1

I=PTR/100

Therefore, 750/452 = R Or

Therefore, Rate = 8 ⅓%

Solution 2

For 2 Years, Interest Paid is 750

Therefore, For 1 Year, Interest Paid 750/2 = 375

On 4,500, Interest Paid is 375

Therefore, On 100, Rate of Interest Paid ==8 ⅓%