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]