Informative line

Rate

• There are many times when we hear the word 'rate'.
• Till now, we have not learned the concept of rate.
• Let's learn it.

Rate

• Rate describes a ratio relationship between two different types of units.

For example: $$1.75$$ miles/hour is a rate that describes a ratio relationship between hours and miles.

If an object travels at this rate then at the end of 1st hour, it will cover $$1.75$$ miles, after 2 hours it will cover $$3.50$$ miles, after 3 hours it will cover $$5.25$$ miles, and so on.

• A rate can be identified by the keyword 'per'.

At a store, $$10$$ boxes contain $$3000$$ chocolates. Which one of the following options represents the rate at which chocolates are packed in the boxes?

A $$\text{30 chocolates}$$

B $$\text{3000 chocolates per 10 boxes}$$

C $$\text{10 boxes}$$

D $$\text{300 chocolates}$$

×

A rate can be identified by the keyword 'per'.

Only option (B) shows a rate, i.e. "$$\text{3000 chocolates per 10 boxes}$$".

Hence, option (B) is correct.

At a store, $$10$$ boxes contain $$3000$$ chocolates. Which one of the following options represents the rate at which chocolates are packed in the boxes?

A

$$\text{30 chocolates}$$

.

B

$$\text{3000 chocolates per 10 boxes}$$

C

$$\text{10 boxes}$$

D

$$\text{300 chocolates}$$

Option B is Correct

Speed

• Speed is the rate at which something moves.
• Speed is the rate of motion or the rate of change of position. It is expressed as distance moved per unit of time.

$$Speed=\dfrac{\text{Distance}}{\text{Time}}$$

For example: Carl travels $$40$$ miles in $$20$$ minutes. Find the distance Carl travels in $$50$$ minutes.

Given statement:

Carl travels $$40$$ miles in $$20$$ minutes.

Thus, the rate is $$40$$ miles per $$20$$ minutes.

Now, to find the unit rate, we need to make the denominator $$1$$.

The GCF of $$40$$ and $$20$$ is $$20$$.

So, divide both $$40$$ and $$20$$ by $$20$$.

$$\Rightarrow\dfrac{40\div20}{20\div20}$$

$$\Rightarrow\dfrac{2}{1}$$

This means Carl travels $$2$$ miles in $$1$$minute.

Now, to find the distance traveled in $$50$$ minutes, we have to multiply both numerator and denominator by $$50$$.

So,

$$\Rightarrow\dfrac{2\times50}{1\times50}$$

$$\Rightarrow\dfrac{100}{50}$$

This means Carl travels $$100$$ miles in $$50$$ minutes.

A man covers a distance of $$45$$ miles in $$5$$ hours. Find the distance he covers in $$10$$ hours.

A $$10\; miles$$

B $$90\; miles$$

C $$70\; miles$$

D $$50\; miles$$

×

Given statement:

Man covers $$45$$ miles in $$5$$ hours.

So, the rate at which he travels is $$45$$ miles/$$5$$ hours = $$\dfrac{45\;miles}{5\;hours}$$

Now, to find the unit rate, we need to make the denominator $$1$$.

The GCF of $$45$$ and $$5$$ is $$5$$.

$$\therefore$$ Divide both $$45$$ and $$5$$ by $$5$$.

$$\Rightarrow\dfrac{45\div5}{5\div5}$$

$$\Rightarrow\dfrac{9}{1}$$

This means he covers $$9$$ miles in $$1$$ hour.

To find the distance he covers in $$10$$ hours, we have to multiply both numerator and denominator by $$10$$.

So,

$$\Rightarrow\dfrac{9\times10}{1\times10}$$

$$\Rightarrow\dfrac{90\; miles}{10\;hours}$$

This means he covers $$90$$ miles in $$10$$ hours.

Hence, option (B) is correct.

A man covers a distance of $$45$$ miles in $$5$$ hours. Find the distance he covers in $$10$$ hours.

A

$$10\; miles$$

.

B

$$90\; miles$$

C

$$70\; miles$$

D

$$50\; miles$$

Option B is Correct

Unit Rate Method

Unit Rate

• A unit rate describes a ratio relationship between two types of quantities, when one of them is $$1$$.
• It describes a ratio in which the denominator is $$1$$.

For example: If an airplane travels at a speed of 610 miles/hr then

$$Unit \,rate=\dfrac{610 \; miles}{1\; hour}$$

Unit Rate Method

• Unit Rate Method is a method in which we first find the value of one unit, and then calculate the value of the required number of units.

For example:

Casey makes $$60$$ cookies in $$3$$ hours.

• To write this in ratio, we need to compare $$60$$ cookies with $$3$$ hours. The $$60$$ cookies become our numerator and $$3$$ hours become our denominator.

Thus, $$\text{rate}=\dfrac{\text{60 cookies}}{\text{3 hours}}$$

Now, to find the unit rate, we have to make the denominator $$1$$.

The GCF of $$60$$ and $$3$$ is $$3$$.

$$\therefore$$ Divide both $$60$$ and $$3$$ by $$3$$.

$$\Rightarrow\dfrac{60\div3}{3\div3}$$

So, we get

$$\Rightarrow \dfrac{\text{20 cookies}}{\text{1 hour}}$$

This means Casey makes $$20$$ cookies per hour.

Alex writes $$120$$ pages in $$2$$ hours. Find the unit rate using unit rate method.

A $$\text{50 pages in 2 hours}$$

B $$\text{60 pages in 2 hours}$$

C $$\text{60 pages in 1 hour}$$

D $$\text{80 pages in 1 hour}$$

×

Given statement:

Alex writes $$120$$ pages in $$2$$ hours.

Thus, we compare $$120$$ pages with $$2$$ hours.

So, we can write,

$$\text{Rate}=\dfrac{\text{120 pages}}{\text{2 hours}}$$

Now, to find the unit rate, we need to make the denominator $$1$$.

The GCF of $$120$$ and $$2$$ is $$2$$.

$$\therefore$$ Divide both $$120$$ and $$2$$ by $$2$$.

$$\Rightarrow\dfrac{120\div2}{2\div2}$$

So, we get

$$\Rightarrow\dfrac{60}{1}$$

This means Alex writes $$60$$ pages in $$1$$ hour.

Hence, option (C) is correct.

Alex writes $$120$$ pages in $$2$$ hours. Find the unit rate using unit rate method.

A

$$\text{50 pages in 2 hours}$$

.

B

$$\text{60 pages in 2 hours}$$

C

$$\text{60 pages in 1 hour}$$

D

$$\text{80 pages in 1 hour}$$

Option C is Correct

Unit Rate or Equivalent Rate

Rate

• Rate describes a ratio relationship between two types of quantities.

For example: Jose can run $$3$$ miles in $$2$$ hours.

Here, $$3$$ miles in $$2$$ hours is a rate that describes a ratio relationship between hours and miles.

• Rate can be identified by the keyword 'per'.

Unit Rate

• A unit rate describes a ratio relationship between two types of quantities, when one of them is $$1$$.

For example: Jose can run $$1.5$$ miles per (1) hour.

Here, $$1.5$$ miles per hour is a unit rate that describes a ratio relationship between miles and hour.

Note: Here, the unit 'miles/hour' is known as rate unit.

Equivalent Rate

• When two or more rates are equal, they are called equivalent rates.

For example: Jose can run $$3$$ miles in $$2$$ hours.

$$\text{Rate}=\dfrac{3\; miles }{2\;hours}$$

$$Unit\, Rate=\dfrac{1.5\;miles}{1\; hour}$$  [By long division method]

Alex can run $$1.5$$ miles per hour.

$$\text{Unit Rate}=\dfrac{1.5\;miles}{1\;hour}$$

Thus, both rates are equal.

$$\dfrac{3\; miles }{2\;hours}$$ $$=\dfrac{1.5\;miles}{1\;hour}$$

These are equivalent rates.

Which one of the following represents the equivalent rate of $$24$$ pastries per $$4$$ boxes?

A $$\text{6 pastries per box}$$

B $$\text{12 pastries per box}$$

C $$\text{1 pastry per box}$$

D $$\text{3 pastries per box}$$

×

Given rate:

$$24$$ pastries per $$4$$ boxes

Its equivalent rate can be calculated by converting it into its unit rate.

Rate $$=\dfrac{24\;pastries}{4\;boxes}$$

We will make denominator $$1$$ by long division method,

$$\Rightarrow\dfrac{6\;pastries}{1 \;box}$$

Thus, the unit rate is $$6$$ pastries per box.

Hence, option (A) is correct.

Which one of the following represents the equivalent rate of $$24$$ pastries per $$4$$ boxes?

A

$$\text{6 pastries per box}$$

.

B

$$\text{12 pastries per box}$$

C

$$\text{1 pastry per box}$$

D

$$\text{3 pastries per box}$$

Option A is Correct

Cost

• Cost is the amount that has to be paid or spent to buy or make something.
• If we know the cost of one unit, we can find out the cost of more units.
• Generally, the unit of cost is dollars(\$).

For example: If the cost of $$5$$ chocolates is $$10$$, find the cost of $$12$$ chocolates.

Cost of $$5$$ chocolates $$=10$$

Thus, the cost of $$1$$ chocolate = $$\dfrac{10}{5}$$

Now, to find the unit rate, we need to make the denominator $$1$$.

The GCF of $$10$$ and  $$5$$ is $$5$$.

$$\therefore$$ Divide both $$10$$ and  $$5$$ by $$5$$.

$$\Rightarrow\dfrac{10\div5}{5\div5}$$

$$\Rightarrow\dfrac{2}{1}$$

This means the cost of $$1$$ chocolate is $$2$$.

Now, to find the cost of $$12$$ chocolates, we have to multiply both numerator and denominator by $$12$$.

So,

$$\text{Rate}=\dfrac{2\times12}{1\times12}$$

$$\Rightarrow\dfrac{24}{12}$$

This means the cost of $$12$$ chocolates is $$24$$.

If the cost of $$10$$ muffins is $$20$$, find the cost of $$30$$ muffins.

A $$20$$

B $$30$$

C $$40$$

D $$60$$

×

Given statement:

Cost of $$10$$ muffins $$=20$$

Cost of $$1$$ muffin $$=\dfrac{20}{10}$$

To find the unit rate, we need to make the denominator $$1$$.

The GCF of  $$20$$ and $$10$$  is $$10$$

$$\therefore$$ Divide both $$20$$ and $$10$$ by $$10$$.

$$\Rightarrow\dfrac{20\div10}{10\div10}$$

$$\Rightarrow\dfrac{2}{1}$$

This means the cost of $$1$$ muffin is $$2$$.

Now, to find the cost of $$30$$ muffins, we have to multiply both numerator and denominator by $$30$$.

$$\Rightarrow\dfrac{2\times30}{1\times30}$$

$$\Rightarrow\dfrac{60}{30}$$

This means the cost of $$30$$ muffins is $$60$$.

Hence, option (D) is correct.

If the cost of $$10$$ muffins is $$20$$, find the cost of $$30$$ muffins.

A

$$20$$

.

B

$$30$$

C

$$40$$

D

$$60$$

Option D is Correct

Time - Work

• Time and work are related to each other.
• Time - work relationship shows how much work is done in a given period of time. When amount of work increases, the amount of time required to do that work also increases.

For example: Jacob writes $$20$$ pages of notes in $$5$$ hours.

Let's calculate the number of pages he can write in $$8$$ hours.

The time taken by Jacob to write $$20$$ pages of notes $$=5$$ hours

$$\text{Rate}=\dfrac{\text{20 pages}}{\text{5 hours}}$$

To find the unit rate, we need to make the denominator $$1$$.

The GCF of $$20$$ and $$5$$ is $$5$$.

$$\therefore$$ Divide both $$20$$ and $$5$$ by $$5$$.

$$\Rightarrow\dfrac{20\div5}{5\div5}$$

$$\Rightarrow\dfrac{4}{1}$$

This means Jacob writes $$4$$ pages per hour.

Now, to find the number of pages he can write in $$8$$ hours, we have to multiply both numerator and denominator by $$8$$.

So,

$$\Rightarrow\dfrac{4\times8}{1\times8}$$

$$\Rightarrow\dfrac{32}{8}$$

This means Jacob writes $$32$$ pages in $$8$$ hours.

Olivia prepares $$40$$ kg of dough in $$10$$ hours. How much dough can she prepare in $$12$$ hours?

A $$90\;kg$$

B $$48\;kg$$

C $$81\;kg$$

D $$10\;kg$$

×

Given statement:

Olivia prepares $$40$$ kg of dough in $$10$$ hours.

So, we can write,

$$\text{Rate}=\dfrac{40}{10}$$

To find the unit rate, we need to make the denominator $$1$$.

The GCF of $$40$$ and $$10$$ is $$10$$.

$$\therefore$$ Divide both $$40$$ and $$10$$ by $$10$$.

$$\Rightarrow\dfrac{40\div10}{10\div10}$$

$$\Rightarrow\dfrac{4}{1}$$

This means she prepares $$4$$ kg of dough per hour.

Now, to find the quantity of dough she can prepare in $$12$$ hours, we have to multiply both numerator and denominator by $$12$$.

So,

$$\Rightarrow\dfrac{4\times12}{1\times12}$$

$$\Rightarrow\dfrac{48}{12}$$

This means she can prepare $$48$$ kg of dough in $$12$$ hours.

Hence, option (B) is correct.

Olivia prepares $$40$$ kg of dough in $$10$$ hours. How much dough can she prepare in $$12$$ hours?

A

$$90\;kg$$

.

B

$$48\;kg$$

C

$$81\;kg$$

D

$$10\;kg$$

Option B is Correct

Comparison of Unit Rates

Rate

• Rate describes a ratio relationship between two types of quantities.

Unit rate

• A unit rate describes a ratio relationship between two types of quantities, when one of them is $$1$$.

Comparison of unit rates

The comparison of unit rates can be a good way of finding out which is the best option/best buy.

Consider the following example:

Casey prepares $$50$$ kgs of dough in $$10$$ hours and Jana prepares $$30$$ kgs of dough in $$15$$ hours. Find who prepares more dough in less time.

Given statement:

Casey prepares $$50$$ kgs of dough in $$10$$ hours.

Thus, we compare $$50$$ kgs with $$10$$ hours.

So, we can write,

$$\Rightarrow\dfrac{50\;kgs}{10\;hours}$$

Now, to find the unit rate, we need to make the denominator 1.

The GCF of $$50$$ and $$10$$ is $$10$$.

$$\therefore$$ Divide both $$50$$ and $$10$$ by $$10$$.

$$\Rightarrow\dfrac{50\div10}{10\div10}$$

So, we get

$$\Rightarrow 5:1$$

$$\Rightarrow\dfrac{5}{1}$$

This means Casey prepares $$5$$ kg dough per hour.

Given statement:

Jana prepares $$30$$ kgs of dough in $$15$$ hours.

Thus, we compare $$30$$ kgs with $$15$$ hours.

So, we can write,

$$\Rightarrow\dfrac{30\;kgs}{15\;hours}$$

Now, to find the unit rate, we need to make the denominator 1.

The GCF of $$30$$ and $$15$$ is $$15$$.

$$\therefore$$ Divide both $$30$$ and $$15$$ by $$15$$.

$$\Rightarrow\dfrac{30\div15}{15\div15}$$

So, we get

$$\Rightarrow 2:1$$

$$\Rightarrow\dfrac{2}{1}$$

This means Jana prepares $$2$$ kg dough per hour.

Thus, Casey prepares more dough in less time as compared to Jana.

Carl earns $$8$$ by selling $$2$$ packets of pens and Jacob earns $$20$$ by selling $$4$$ packets of similar pens. Find who earns less amount on selling $$1$$ packet of pens.

A Carl

B Jacob

C Both earn equal amount

D

×

Given statement:

Carl earns $$8$$ by selling $$2$$ packets of pens.

Thus, we compare $$8$$ with $$2$$ packets.

So, we can write,

$$\Rightarrow\dfrac{8}{2\;packets}$$

Now, to find the unit rate, we need to make the denominator 1.

The GCF of $$8$$ and $$2$$ is $$2$$.

$$\therefore$$ Divide both $$8$$ and $$2$$ by $$2$$.

$$\Rightarrow\dfrac{8\div2}{2\div2}$$

So, we get

$$\Rightarrow4:1$$

$$\Rightarrow\dfrac{4}{1}$$

This means Carl earns $$4$$ per packet.

Given statement:

Jacob earns $$20$$ by selling $$4$$ packets of pens.

Thus, we compare $$20$$ with $$4$$ packets.

So, we can write,

$$\Rightarrow \dfrac{20}{4\;packets}$$

Now, to find the unit rate, we need to make the denominator 1.

The GCF of $$20$$ and $$4$$ is $$4$$.

$$\therefore$$ Divide both $$20$$ and $$4$$ by $$4$$.

$$\Rightarrow\dfrac{20\div4}{4\div4}$$

So, we get

$$\Rightarrow5:1$$

$$\Rightarrow\dfrac{5}{1}$$

This means Jacob earns $$5$$ per packet.

Thus, Carl earns less amount as compared to Jacob on selling $$1$$ packet of pens.

Hence, option (A) is correct.

Carl earns $$8$$ by selling $$2$$ packets of pens and Jacob earns $$20$$ by selling $$4$$ packets of similar pens. Find who earns less amount on selling $$1$$ packet of pens.

A

Carl

.

B

Jacob

C

Both earn equal amount

D

Option A is Correct