Applications of Ratio and Proportion

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الشرح

Scale drawing

A scale drawing represents a real object using a fixed ratio between every drawing length and the matching real length:

\text{scale}=\frac{\text{length in drawing}}{\text{real length}}

A scale drawing is an accurate smaller copy of the real object.

A scale of $1:600000$ means $1$ cm on the map stands for $600000$ cm (that is $6$ km) in reality. The two numbers of a scale must be in the same unit before you compare them.

Rate and unit rate

A rate compares two quantities of different kinds, such as

\frac{120\text{ km}}{2\text{ h}}

. A unit rate has

1

in the denominator, e.g.

60

km per

1

Sharing in a ratio

To divide an amount in the ratio

a:b

, split it into

a+b

equal parts, then give

a

parts to the first share and

b

\text{real length}=\frac{\text{drawing length}}{\text{scale}}

Rearranging the scale relationship to recover a real length.

Percentage applications

A percentage is a ratio out of $100$ , so $r\%=\frac{r}{100}$ . Percentages let us compare and change prices in everyday life: discounts, taxes and profit or loss.

Discount and sale price

When a price is reduced by a discount rate

d\%

, the discount (deduction) value

=\text{price}\times\frac{d}{100}

, and the sale price

=\text{price}-\text{discount}

Adding sales tax (VAT)

A sales tax adds a rate

t\%

to a price: the tax value

=\text{price}\times\frac{t}{100}

, and the amount paid

=\text{price}+\text{tax value}

Profit and loss

If an item is bought for a cost price and sold for a selling price, then a profit

=\text{selling}-\text{cost}

(when selling

>

cost) or a loss

=\text{cost}-\text{selling}

. The profit or loss percentage is always taken of the cost price:

\text{profit}\%=\frac{\text{profit}}{\text{cost}}\times100

The whole amount = 60.

▶ Watch a total being shared in a given ratio.

أمثلة محلولة

Worked exampleUsing a map scale

A map has scale $1:600000$ . Two towns are $4.5$ cm apart on the map. Find the real distance in kilometres.

Worked exampleSharing in a ratio (two parts)

Share $60$ LE between two friends in the ratio $2:3$ .

Worked exampleReal length to drawing length

A wall is $6$ m long. On a plan with scale $1:200$ , how long is the wall in centimetres?

Worked exampleComparing unit rates (best value)

Shop A sells $5$ pens for $20$ LE; shop B sells $8$ pens for $28$ LE. Which shop is cheaper per pen?

Worked exampleSharing in a ratio (three parts)

Divide $120$ LE among three children in the ratio $1:2:3$ .

Worked exampleFinding the total from one share

Two partners share a profit in the ratio $3:5$ . The first partner receives $600$ LE. Find the total profit.

Worked examplePrice after a discount

A smart phone is priced at $12750$ LE. A shop offers an $8\%$ discount. Find the sale price.

Worked exampleAdding sales tax

A home Internet service costs $520$ LE per month, plus a $14\%$ sales tax. Find the amount actually paid each month.