Total oranges shared = 20
Ama's share = 12
Araba's share = 20 - 12
Araba's share = 8

Araba:Ama = 8:12
Araba:Ama = 2:3

Note: 4 divides 8, 2 times and 12, 3 times.

If $\frac{5}{8}$ = 1 kilometre
240 mile = 1 kilometre x 240 mile ÷ $\frac{5}{8}$ mile

Change the division sign to multiplication and reciprocate the fraction.

240 mile = 1 kilometre x 240 x $\frac{8}{5}$

240 mile = 1 kilometer x 48 x 8

5 divides itself once and 240, 48 times.

240 mile = 384 kilometres

Dividing 240 by 5

The more the workers, the lesser the time to complete the task. Hence number of workers is inversely proportional to time.

Let w = workers, t = time to complete task

w ∝ $\frac{1}{\mathrm{t}}$

We can introduce a constant to remove the proportionality.

Let k = proportionality constant.

w = k x $\frac{1}{\mathrm{t}}$

w = $\frac{\mathrm{k}}{\mathrm{t}}$

Make k the subject.

Multiply both sides by t

k = w x t

The value of k is the same for all scenarios since they work at the same rate.

k value for 1 man is the same for the 6 men.

Let t = time taken for the 6 men to complete the task.

1 x 12 = 6 x t
Divide both sides by 6 so that t stands alone

t = $\frac{\mathrm{1\; x\; 12}}{6}$ = $\frac{12}{6}$ = 2 hours

It will take 2 hours for the 6 men to complete the work at the same rate.

If 1 m = 100 cm
12.03 m = $\frac{\mathrm{12.03\; m\; x\; 100\; cm}}{\mathrm{1\; m}}$ = 1203 cm

Note: When multiplying a decimal number by 1n number of zeros, you move the decimal point to the right n times.

12.03 x 100 = 12.⌒0⌒3. = 1203

Cement:Sand = 2:5

Total ratios = 2 + 5 = 7

Ratio for cement = 2

If 7 = 35 kg
2 = $\frac{\mathrm{2\; x\; 35\; kg}}{7}$ = 2 x 5 kg = 10 kg

NOTE: since the duration was given in years, you have to convert the years to months before multiplying the cost per month

There are 12 months in a year

Half a year is 6 months

Hence 1½ years = 12 months + 6 months = 18 months

Method I

Using proportionality.

Per month means every (1) month.

If 1 month = GH₵ 9

18 months = $\frac{\mathrm{GH₵\; 9\; x\; 18\; months}}{\mathrm{1\; month}}$ = GH₵ 9 x 18 = GH₵ 162.00

Method II

Total Amount paid = Total Months x Cost per month

Total amount paid = 18 months x GH₵ 9.00 per month

Total amount paid = 18 x 9 = GH₵ 162.00

Since the first part of the ratio 1 : n is 1, divide each parts of the ratio of 15 : 12 by 15 (15/15 = 1)

15/15 : 12/15 = 1:0.8

13 for farmers and 11 for children

Farmers = 312

Let x = number of children

13 = 312

11 = x

Cross multiply and solve for the value of x

13 x x = 312 x 11

Divide both sides by 13

x = (312 x 11)/13

13 goes into 312, 24 times

x = 24 x 11

x = 264

Kofi’s ratio is 2 and Ama’s ratio is 3 from the Kofi : Ama = 2:3

Sum of ratios = 2 + 3 = 5

NOTE: 5 goes into 30, 6 times then you multiply 2 by 600 to get the 1,200

ASSIGNMENT

Calculate the amount received by Ama

SOLUTION

Ama received 1,800. Alternatively you can subtract 1200 (Amount received by Kofi) from 3000 (Total amount shared) to get Ama’s amount.

1⁄3 = 5 km

18 cm = ?

Let 18 cm = xkm and cross multiply to solve for the value of x

1⁄3 x x = 5 x 18

Applying the principles of multiplying fractions

Every number is being divided by 1 hence x = x/1

1⁄3 x x = 5 x 18

Get rid of the fraction by multiplying both sides of the equation by the L.C.M of the denominators.

The L.C.M of 3 and 1 is 3, hence multiply both sides of the equation by 3

x = 270

Alternatively

You can cross multiply and solve for the value of x

x x 1 = 90 x 3

x = 270

Alternatively, you can calculate how many times 1⁄3 can go into 18 and multiply the result by 5 km

Applying the principles of dividing fractions

Thus when dividing fractions, you reverse the fraction at the right and change the division to multiplication

Now you can use the principles of multiplying fractions to simplify

Every number is being divided by 1

12 goes into 156, 13 times

Number of boxes = 13

ALTERNATIVELY

Let x = Number of boxes

12 pencils = 1 box

156 pencils = x box

Cross multiply to solve for the value of x

12 x x = 156 x 1

Divide both sides by 12

12 goes into 156, 13 times

Number of boxes = 13

ALTERNATIVELY

Using ratio and proportion

If 12 pencils = 1 box

156 pencils = $\frac{\mathrm{1\; box\; x\; 156\; pencils}}{\mathrm{12\; pencils}}$ = 13 boxes

Note:
1. Pencils cancel each other
2. 12 divides itself 1 time and 156, 13 times

Method I

The sum of ratios represent the total amount shared

The difference (larger ratio - smaller ratio) represents how much more

Zalia : Amina = 2 : 5

Sum of ratios = 2 + 5 = 7

Difference of ratios = 5 - 2 = 3

Amina had 150 more than Zalia.

If 3 = 150

7 = $\frac{\mathrm{7\; x\; 150}}{3}$ = 7 x 50 = 350

Total amount shared = GH₵ 350

Method II

Let x = Zalia's share

Amina had GH₵ 150.00 more than Zalia

Amina's share = Zalia's share + GH₵ 150

Amina's share = x + 150

Zalia share : Amina share = 2 : 5

x : x + 150 = 2 : 5

Change the ratios to fractions.

$\frac{\mathrm{x}}{\mathrm{x\; +\; 150}}$ = $\frac{2}{5}$

Cross multiply

5 x x = 2 x (x + 150)

5x = 2x + 300

5x - 2x = 300

3x = 300

Divide both sides by 3

x = $\frac{300}{3}$

x = 100

∴ Zalia's share = GH₵ 100.00

Amina's share = Zalia's share + GH₵ 150.00

Amina's share = GH₵ 100.00 + GH₵ 150.00

Amina's share = GH₵ 250.00

Total amount shared = Zalia's share + Amina's share

Total amount shared = GH₵ 100.00 + GH₵ 250.00

Total amount shared = GH₵ 350

Let x be the cost of one book

12 books = 54.84

1 book = x

Cross multiplying the equations

⇒ 12 x x = 1 x 54.84

Divide both sides by 12

⇒ x = 54.84 / 12

⇒ x = 4.57

Thus

To simplify the fraction, change the decimal number to a form which has 10 raised to the power by moving the decimal position to the right until you get a whole number.

Multiply that whole number by 10 raised to the power negative of the number of times you moved to the right

Thus 58.84 = 5884 x 10^-2

NOTE:The power is negative if you moved to the right and positive if you move to the left

⇒ x = 54.84 / 12 = 5884 x 10^-2/12

Using long division to simplify,

⇒ 5884 x 10^-2/12 = 457 x 10^-2

Now change 457 x 10^-2 back to decimal.

457 is the same as 457.0, thus every whole number has point zero decimal

Since the 10 is raised to the power -2, we move the decimal point to the left 2 times, which gives us 4.57

Alternatively, you can multiply each of the possible answers by 12 and the one that gives you the total cost of the 12 books (GH₵ 54.84) is the answer.

Each person's share from the ratio is calculated as:
(The ratio / Sum of ratios) x Total being shared

Vera:Brother = 11:9

Sum of ratios = 11+9 = 20

Vera's share = (11/20) x 60 = 11 x 3 = 33

NOTE: 20 goes into 60, 3 times and the 3 multiplies the 11 at the top and the result is 33

Brother's share = (9/20) x 60 = 9 x 3 = 27

To get how much more Vera got, subtract the brother's share from Vera's share to know how much more she got

Thus 33 - 27 = 6

Total wage = Total hours worked x wage per hour

½ = 0.5 ⇒ 20½ = 20.5

wage per hour = 2.5

⇒ Total wage = 20.5 x 2.5 = 51.25

Percentage = Value/Total x 100

Value here is the number of boys (480)

The percentage is 80

Let x = total number of pupils in the school

⇒ 80 = 480/x x 100

Get rid of the x dividing (fraction) by multiplying both sides of the equation by x

⇒ 80 x x = 480 x 100

Divide both sides of the equation by 80

⇒ x = 480 x 100/80

80 will go into 480, 6 times

⇒ x = 480 x 100/80 = 6 x 100 = 600

Hence there are 600 pupils in the school

Members in the club = No. of females + No. of males

Concept

No. of A : No. of B = Ratio of A:B

⇒ No. of females : No. of males = 3:2

No. of females = 15

⇒ 15: No. of males = 3:2

Express the : as fractions

⇒ 15/No. of males = 3/2

Cross multiply (The numerator from one side multiplies the denominator from the other side of the equation)

No. of males x 3 = 15 x 2

Make No. of males the subject by dividing both sides of the equation by 3

No. of males = 15 x 2 / 3

3 goes into 15, 5 times

⇒ No. of males = 5 x 2 = 10

Members in the club = No. of females + No. of males

⇒ Members in the club = 15+10 = 25