TheMathBooklets

Monday, 20 January 2020

N2-G123 Ratios

Ratio compares two or more quantities.

Symbol of ratio is :

Ratio of quantity A to quantity B s written as

A : B

** Ratio has no unit of measure

Example

Write the ratio 200: 250 in its simplest form

200 : 250

20÷5 : 25 ÷ 5

4 : 5

a : b: c => a: b , b : c

Equivalent Ratios

Example

4 : 10

= 4 _÷2 : 10_÷2

= 2 : 5

4 : 10 and 2 : 5 are equivalent ratios.

Ratio in its simplest form

Example

Write 8 : 12: 36 in its simplest form

Step: Find the common factors for the given numbers and reduce

8 _÷4 : 12_÷4 : 36_÷4

2 : 3 : 9

Example: Write the ratio 200: 250 in its simplest form

200 : 250

20÷5 : 25 ÷ 5

4 : 5

Example

The ratio of the number of tables to the number of chairs is 1 : 4 in the restaurant. There are 6 tables, how many chairs are there?

Step 1 : Write the ratio and number of tables

T C

1 : 4

Step 2 : Write the number of tables on both side of the ratio and multiply

T C

1 : 4

x 4 x4

= 4 : 16

Step3 : Answer

There are 16 chairs

Ratio with different Unit of Measure (UOM)

The UOM must be the same when comparing two of more quantities

Example:

John has $1 and Sally has 80cts as their daily allowance. What is the ratio of their allowance?

Step 1 : Convert the amount to the same unit (of the 'smaller' UOM)

$1 = 100cts

Step 2 : Write the ratio and reduce the the lowest term

100 : 80

₅ 10 : 8 ₄

5 : 4

Step 3 : Write the Answer

The ration is 5 : 4

Ratio involving Fractions and Decimals

Example

Express the following ratios in its simplest form.

1.2 : 3

Step 1 : Multiply 10 to both side (to remove the decimal)

x10. 1.2. : 3 x 10

12 : 30

Step 2 : Reduce the simplest term

<< divide by 6 for both side >>

2 12 : 30 5

The ratio is 2 : 5

Example

Express the following ratios in its simplest form.

b. 1 ¾. : 2 ½

Step 1: Change both to improper fractions

7 : 5

4 2

Step 2 : Make both side to have the same denominator

<< multiply 2 , by 2 >>

7 : 5 x 2

4 2 x 2

7 : 10

4 4

Step 3 : Write the numerators as the ratio

7 : 10

Connecting Ratio with Fraction

For a ratio A : B, the 'total number of parts' = A + B

Thus in fraction form, A's part is A / A+B

B's part is B / A+B

Example

There are 10 girls and 15 boys.

The ratio of the number of girls to the number of boys =

10 : 15 or 2 : 3

The total part = 2 + 3 = 5

=> Number of girls = 2/3 of number of boys

=> Number of girls = 2/5 of the total

=> Number of boys = 3/2 of number of girls

=> Number of boys = 3/5 of the total.

Example

Abby and Bobby have 10 oranges altogether.

The oranges are to be divided in the ratio of 3 : 2

How many oranges will each of them get?

Step 1: Find the total parts

Total = 3 + 2 = 5

Step 2 : Compute their fraction's

Abby = 3/5 , Bobby = 2/5

Step 3 : Compute by multiplying total quantity with the fractions

Abby has 3/5 x 10 = 6 oranges

Bobby has 2/5 x 10 = 4 oranges

Practice

1. Express the following ratios in its simplest form.

(a) 4 : 20 [1 : 5]

(b) 35 : 14 [7 : 2]

2. Express the following ratios in its simplest form.

a. ⅚ kg : 2 kg [5 : 12 ]

b. 1.6 cm : 4 mm [4 : 1 ]

c. 1 ½ hr : 20 mins [9 : 2 ]

d. 250 ml : 1 litre [1 : 4 ]

3. Mary and Andy share a pizza in the ratio of 1 : 2. Andy gets 4 slices of the pizza. How many slices does Mary get? [2]

4. Alice, Betty and Coby share their marbles in the ratio of 2 : 3 : 5. Find the amount each gets for each of the marbles.

a. 420 (b) 70 [a. 84, 126, 210 b. 14, 21, 35]

N3-G123 Percentage

Percentage (S1 - NT)

A percentage is expressed as a number our of 100.

Symbol: %

% means out of 100

25% => 25/100

% and fraction

Example: Express 20% as a fraction

Step : Change % to /100 and reduce to lowest term

20% => 20/100

= 1/5

Example: Express 1/4 as a percentage

Step: Multiply by 100 and add a percent

1/4 = 1/4 x 100

= 25%

% and Decimal

Example: Express 25% as a decimal

Step: Divide number by 100 and remove the %

25% = 25/100

= 0.25

Example: Express 0.38 as a percentage

Step : Multiply by 100 and add %

38% = 38/100

= 0.38

Percentage greater than 100%

100% = 100/100 = 1

Example:

Express 120% as (a) a decimal (b) a fraction

Step 1: Write the percent with /100

120% = 120/100

Step 2: 'Split' value with 100/100 or 1

120/100 = 1 20/100

Step 3: Compute required value

(a) 20/100 = 1/5

120% = 1 1/5

(b) 20/100 = 0.2

120% = 1.2

Finding a Percentage Part of a Whole

=> To Find a Quantity Given the Percentage

To Find the value of A% of a quantity B,

=> Changing the A% to /100

A x B

100

Example

Calculate 25% of 80

Step1: Use the formula A/100 x B

25/100 x 80

Step2: Compute the value

25/100₄₁ x 80²⁰

= 20

Example: What is 6% of 300?

6% of 300

= 6/100 x 300 (step 1)

= 18 (step 2)

Example: What is 110% of 50?

110% of 50 = 110/100 x 50 (Step 1)

= 55 (Step 2)

Expressing a Quantity as a percentage

Example

Cindy has $20. She spent $15 to buy a dress. What percentage of the money is spent?

She has $20 (Whole). She spent $15 (part).

Expressing A(part) as a percentage of B(whole) :

=> A x 100%

Step1: Use formula

= 15/20 x 100

Step 2 : Compute the value

= 3/4 x 100

= 75%

Example

Express 9g as a percent of 150g

= 9/150 x 100. (Step 1)

= 9₃ x 10₂ 0 (Step 2)

15₅0

= 3 x 2

= 6%

Finding the Whole Given a Percentage

Example

Anna spends 10% of her money to buy 1 pencil. If the pen costs $2. How much money does she have at first (100%)?

Step1: Relate the % to the value

10% -> $2

Step2: Find 1% of the value

1% -> 2/10 = $0.2

Step3: Find the whole (100%)

100% -> 100 x 0.2

= $20

Comparing Two Quantities by Percentage

Example

Gina scored 7/10 in her first Science test, and 16/25 for her second test. Which test did she score better?

Step 1: Convert both marks to %

7/10 = 7/10 x 100 = 70%

16/25 = 17/25 x 100 = 68%

Step 2 : Compare the two percent value and Answer

She did better in her first test.

Increasing/Decreasing a Quantity by a Given Percentage

Example

Increase $500 by 20%

Step1: Find 1% of $500

1% -> 500/100 = $5

Step2: Compute required percent or Step2: total of percent increase

20% -> 20 x 5 120% -> 120 x 5 = $600

= $100

Step3: (Increase) Add to original Amount

Increase = $500 + $100 = $600

Finding Percentage Change (Increase/Decrease)

Example

John's pay increased from $2000 to $2000. How many percent has his pay increased?

Step1: Compute the increase/decrease

$2200 - $2000 = $200

Step2: Amount increased x 100. [Formula]

original amount

= 12/20 x 100 [** the denominator is the Whole(original amount0]

Step3: Compute the value

12₆/20₁₀ x 100

= 60%

=> Formula : Amount Increase/Decreased x 100

Original Amount

Example

Calculate the percentage change for increasing 20 to 32

Increase = 32 – 20. (Step 1)

= 12

% change = 12/20 x 100 (Step 2)

= 12₆/20₁₀ x 100 (Step 3)

= 60%

Finding the Whole Given the Percentage Increase/Decrease

Example

The price of a book was increased by $2. The increase is equal to 10% of the original price. Find the original price of the book.

10% = $2 (Step 1 : Find 1%)

1% = $0.2

100% = $0.2 x 100 (Step 2 : Find 100%)

= $20.

* When calculating percentage changes (increase or decrease), always divide by the original value.

Profit and Loss

Cost Price = The price to make/buy something to sell

Selling Price = The price to sell something

There is a profit when Selling Price > Cost Price

There is a loss when Cost Price > Selling Price

Profit = Selling Price - Cost Price

=> Selling Price = Cost Price + Profit

Loss = Cost Price - Selling Price

Example

Kathy bought a bag for $25, and sell it for $30.

(a) What is her profit?

Cost Price = $25

Selling Price = $30

Profit = $30 - $25 = $5

(b) What is her profit as a percentage of the cost price?

Profit % = Profit x 100%

Cost Price

Example

A watch is sold at a profit of 20%.

(a) Find the cost price of the bag if the profit is $40.

20% = $40

1% = $40/20 = $2 (Step 1 : Find 1%)

100% = $2 x 100 (Step 2 : Find 100% = Cost Price)

= $200

(b) What is the selling price?

Selling price = Cost price + profit

= $200 + $40 = $240.

Practice

1. Express the following percentages as fractions in the simplest form.

(a) 5% (b) 5 1/2% (c) 1.4% (d) 0.8% (e) 125%

2. Express the following percentages as decimals.

(a)10% (b) 53% (c) 180% (d) 0.75% (e) 6.5%

3. Express each of the following as a percentage

(a) 1/2

(b) 0.35

(d) 3/5

4. Find the following:

(a) 20% of $45

(b) 80% of $200

4. Express

(a) 21g as a percentage of 70g

(b) 15 mins as a percentage of 1 hour

5. Which has a higher percentage: 40 marks out of 50 or 45 marks out of 60?

6. Calculate the increase or decrease.

(a) Increase the time of 90 minutes by 12%

(b) Decrease the selling of $64 by 25%

7. The new packaging for a packet of chips increased from 300g to 360g. Find the percentage increase.

8. The marked price of a tablet is $267.50 inclusive of GST. If the GST is 7%, find

(a) its price before GST

(b) the amount of GST levied on it.

9. The original price for a pair of shoe is $25. At a sale, it was sold for $22. Find the percent decrease in price.

N22-G23 Proportion

Direct Proportion

Direct proportion: As one variable (Y) increases ↑, another variable (X) increases↑ at the same proportion/rate.

=> as y increase ↑, x also increases ↑ proportionally

y α x Symbol of proportion: α

The equation for direct proportion is

y = kx (where k is the constant of proportionality)

Example y is proportional to x and y = 20 when x = 25. Find y when x = 10.

Step 1: Write out the Formula

y α x

Step 2: Find k (the constant)

y = kx

20 = k x 25

k = 20 / 25 = 4/5

Step 3: Solve

y = 4/5 x

When x = 10,

y = 4/5 x 10

= 8

Inversely Proportion

Inversely: As one variable (Y) increases↑, another variable (X) decreases ↓ at the same proportion/rate.

=> As y increases ↑, x decreases ↓ proportionally

The equation for inverse proportion:

y= k

x (where k is the constant of proportionality)

Example:

y is inversely proportional to x and y = 20 when x = 25. Find y when x = 10.

Step 1: Write the formula

y α 1/x

y = k/x

Step 2: Find k (the constant)

20 = k/25

k = 20 x 25

= 500

Step 3: Solve

y = 500/x

When x = 10,

y = 500/10 = 50

Practice

1. A swimming pool can be filled with water in 12 hours using 4 pumps. How many hours would it take if 8 pumps were used? [17/II/4/2/T]

2. Team A have won 14 of their 20 matches. Team B have won 2/3 of their matches. Which team have won the greater proportion of their matches [16/II/7/3/T]

3. The time taken to build a wall is inversely proportional to the number of people building it. 8 people can build the wall in 15 days.

How long will it take 12 people to build it? [11/II/17a/2/A]

4. Y is directly proportional to x3. When x has a certain value, y = 5. Find the value of y when x is doubled. [11/I/21b/2/A]

5. S is directly proportional to t2. Given that s = 27 when t = 9, find

a. an expression fro s in terms of t.

b. the value of t when s = 1/12 [13/I/14/3/A]

N2-G123 Map Scale

Map Scale (G1-S3)

Scale of Drawing = Drawing length : Actual length

Map Scale = Map distance : Actual Distance

If the scale of a map is given as 1 : 200 000

=> 1cm represents 200 000 cm (200 000 cm = 2000m = 2km)

=> 1cm represents 2 km

1cm : 2km

1cm x 1cm = 1cm²: 2km x 2 km = 4km²

1cm²: 4 km²

* Area calculation: (first calculate 1cm²represent how many km²)

Example

The scale of a map is given as 1 : 200 000

Scale representation and conversion

a. Complete the following

Scale of map is 1cm to km

Step 1: Write the conversion

1km = 100 x 1000

= 100 000 cm

Step 2: Find the conversion scale

Divide given scale by conversion

200 000 / 1001000 = 2km

Step 3: Solve

Scale of map is 1cm to 2 km

To Find Distance on the map ( Given actual distance)

b. Find the distance on the map between two towns that are actually 30 km apart.

Step1: Write the scale

1 cm : 2 km [[ MUST write this scale FIRST when doing ALL questions ]]

Step2: ‘line’ the number and compute

1cm : 2km

? : 30km

30 km apart is 30/2 = 15 cm on map

Distance on the map is 15 cm.

To find Actual Distance (Given distance on the map)

c. Find the actual distance of two towns are that 10 cm apart on the map.

Working:

1 cm : 2 km

10 cm = 10 x 2 = 20

Actual distance = 20 km

To Find Actual Area (Given area of map)

d. A forest has an area of 5 cm²on the map. Find the actual area of the forest in square kilometres.

Step 1: Write the scale and compute to area mapping

1 cm : 2 km

1cm² => 2km x 2km = 4 km²

Step 2: ‘line’ the number and compute (multiply to find actual value)

1cm² : 4 km²

5 cm² : ?

5cm² => 4 x 5 = 20

Actual area of forest = 20 km²

To find Area on Map (Given actual area)

e. The actual area of a town is 120 km2. What is the area on the map?

Step 1: Write the scale and compute to area mapping

1 cm : 2 km

1cm² => 2km x 2km = 4 km²

Step 2: ‘line’ the number and compute (divide to find map value)

1cm² : 4 km²

? : 120 km²

120 km² => 120 / 4 = 30cm²

Area of town on map is 30cm²

Practice

1. Write 480 : 400 in its simplest form.

2. The scale of the map in given as 1cm : 2 km

a. Scale of the map is 1 : ___________

b. The distance between two towns is 30km. What is this distance, in cm on the map?

c. A forest has an area of 8cm² on the map. What is the actual area of the forest in km².

3a. Write 50 000 cm in kilometres.

A map is drawn to a scale of 1 : 50 000

b. The distance between two stations on the map is 12cm

Find the real distance between the stations in kilometres

c. The distance between two towns is 35 km. How far is this distance on the map?