Scaling and Proportions

Scaling is directly related to proportions because all dimensions change in the same ratio.

For example, if the dimensions of a rectangle are doubled:

then the ratios remain the same:

This means the enlarged rectangle has the same shape as the original rectangle.

Manupilating Proportions:

To understand how to manupilate proportions, we start with a simple proof of

then the following are also equal:

where (k) is any constant.

These results follow from the fact that if two ratios are equal, they are both equal to the same constant value.

For example, if:

then:

Adding gives:

which implies:

Similar reasoning works for subtraction and for expressions like:

Key Ideas

• Direct proportion: $y = kx$. Doubling $x$ doubles $y$.
• Inverse proportion: $y = \frac{k}{x}$. Doubling $x$ halves $y$.
• Cross-multiplication: $\frac{a}{b} = \frac{c}{d}$ implies $ad = bc$.
• Scaling a recipe by factor $r$ multiplies every ingredient by $r$.
• Percent increase/decrease can be modeled with multipliers: increase by $p\%$ means multiply by $1+\frac{p}{100}$, decrease by $p\%$ means multiply by $1-\frac{p}{100}$.

Solved Examples:

Example 1:

If x and y are inversely proportional and $x = 10$ when $y = 6$, what is $x$ when $y = 4$?

Solution: We are told $xy = (6)(10) = 60$, so when $y = 4$, we have $xy = 4x = 60$ and $x = 15$.

Example 2:

Given that $x$ is directly proportional to $y$ and to $z$ and is inversely proportional to w,and that $x = 4$ when $(w, y, z) = (6, 8,5)$, what is x when $(w, y, z) = (4,10,9)$?

Solution: Because $x$ is inversely proportional to $w$, when all other variables are constant, $xw$ is constant. Similarly, when the other two variables are constant, each of $x/y$ and $x/z$ is constant. We can combine all these by saying $xw/yz$ is constant. Thus, from the problem we have

Example 3:

It is three o'clock now. How many minutes will pass before the minute and hour hands of a clock are exactly opposite each other?

Solution: At 3 o'clock, the minute hand is at 12 while the hour hand is at 3. Since each hour mark represents 5 minutes on the clock, the hour hand starts 15 minute marks ahead of the minute hand.

For the hands to be opposite each other, they must be separated by 30 minute marks on the clock face.

Suppose (x) minutes have passed.

• The minute hand will be at (x) minute marks.
• The hour hand moves as time passes. In (x) minutes, it moves:

of the distance from one hour mark to the next.

Since one hour mark corresponds to 5 minute marks on the clock, the hour hand moves:

minute marks.

So the hour hand's position is:

We want the distance between the two hands to be 30 minute marks:

Simplifying:

Thus, the hands will be opposite each other after:

or approximately:

Practice Problems:

Example 1

What percent of (20) is (13)?

Example 2

A number is increased by (50%), and then the new number is decreased by (40%). If the final result is (8) less than the original number, what was the original number?

Example 3

A test consists of two sections:

• Section A contributes (60%) of the total grade.
• Section B contributes (40%) of the total grade.

A student scores (95%) on Section A. What exact score must the student obtain on Section B to achieve an overall average of (90%)?

Example 4

A bag contains some Gummy Bears.

• Half of them are given to Jessica.
• One-third of them are given to Jana.
• The remaining (15) are given to Julie.

If the bag is now empty, how many Gummy Bears were originally in the bag?

Example 5

A runner moves at a rate of (x) feet every (y) seconds.

How many yards can the runner cover in (z) minutes?

Example 6

Suppose (y) varies inversely as (x^3).

If:

$$y = 3 \quad \text{when} \quad x = 2$$

find the value of (y) when:

$$x = 9$$

assuming (y > 0).

Example 7

Suppose (x) is directly proportional to (y) and inversely proportional to (z).

If:

$$x = \frac{1}{2} \quad \text{when} \quad y = \frac{3}{4} \quad \text{and} \quad z = \frac{2}{3}$$

find (x) when:

$$y = \frac{7}{8} \quad \text{and} \quad z = \frac{7}{9}$$

Example 8

Four tennis instructors, Lob, Love, Smash, and Vantage, divide their earnings according to the following ratios:

$$\text{Lob} : \text{Love} = 17 : 12$$

$$\text{Love} : \text{Smash} = 3 : 4$$

$$\text{Smash} : \text{Vantage} = 32 : 15$$

If the total earnings are:

$$\$3150$$

how much money does Love receive?

Example 9

Inside a sealed container, the product of pressure and volume remains constant.

If the pressure is increased by (25%), by what percent must the volume decrease?

Example 10

The Hour hand is currently between $10:00$ and $11:00$

Six minutes from now, the minute hand will be exactly opposite the position occupied by the hour hand three minutes ago.

What is the exact current time?

Example 11

A train passes over railroad rails that are each (30) feet long. Each time the train crosses a joint between rails, a click is heard.

The train’s speed in miles per hour is numerically equal to the number of clicks heard in how many seconds?

Example 12

The cost of living rises by (2%) every quarter of the year.

To the nearest tenth of a percent, what annual percentage increase does this correspond to?

Example 13

Two joggers run around the same oval track in opposite directions.

• One jogger completes one lap in (56) seconds.
• The joggers meet every (24) seconds.

How many seconds does the second jogger take to complete one lap?

Example 14

Country A contains (c%) of the world's population and owns (d%) of the world's wealth.

Country B contains (e%) of the world's population and owns (f%) of the world's wealth.

Assuming wealth is distributed equally within each country, find the ratio:

$$\frac{\text{wealth per person in A}}{\text{wealth per person in B}}$$

in terms of (c,d,e,) and (f).

Example 15

For distinct positive numbers (x,y,z),

$$\frac{x+y}{z} = \frac{x}{y} = \frac{y}{x-z}$$

find:

$$\frac{x}{y}$$