Scale Factor As A Fraction, Decimal And A Percentage

📝 Practice Worksheets

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Scale Factor Practice Lab Similarity Explorer

🔑 Key Concepts

The original square has a side length of \(2\) units.
The scale factor is found by dividing the new side length by the original side length.
The rule is \(\displaystyle k=\frac{\text{new}}{\text{original}}\).
A scale factor greater than \(1\) produces an enlargement.
A scale factor between \(0\) and \(1\) produces a reduction.
A scale factor of \(1\) produces a congruent copy with no change in size.
A scale factor may be written as a fraction, decimal, or percent.
For example, \(\frac{3}{2}=1.5=150\%\).
A percent scale factor describes the new size as a percent of the original size.

✏️ Interactive Worked Example

Scale a \(2\times2\) Square

Same Size

Original: \(2\times2\)

New side: 2 units

New side length

2 units

Move the slider in \(0.5\)-unit increments. The blue original square remains \(2\times2\), while the yellow scaled copy changes precisely on the coordinate plane.

Side lengths

Original = 2

New = 2

Scale factor

k = 1

Decimal: 1

Equivalent forms

1 = 1

100%

Transformation

No change

0% change

Step 1: Compare New to Original

The original square has a side length of \(2\) units. The slider controls the new side length.

\(\displaystyle k=\frac{\text{new side length}}{\text{original side length}}\)

\(\displaystyle k=\frac{\text{new}}{2}\)

Both the horizontal and vertical side lengths are multiplied by the same value, so the new figure remains a square and is similar to the original.

Step 2: Interpret the Scale Factor

The value of \(k\) tells whether the new square is an enlargement, reduction, or unchanged copy.

\(0<k<1\quad\Rightarrow\quad\text{reduction}\)

\(k=1\quad\Rightarrow\quad\text{same size}\)

\(k>1\quad\Rightarrow\quad\text{enlargement}\)

Step 3: Connect Fraction, Decimal, and Percent

A scale factor can be expressed in several equivalent forms.

\(\displaystyle \frac{3}{2}=1.5=150\%\)

\(\displaystyle \frac{1}{2}=0.5=50\%\)

\(\displaystyle \frac{5}{2}=2.5=250\%\)

The percent tells how large the new side length is compared with the original side length.

Current Scale-Factor Calculation

Fraction

\(\displaystyle k=\frac{2}{2}=1\)

Divide the new side length by the original side length.

Decimal

\(k=1.0\)

The decimal is the quotient of the new and original side lengths.

Percent

\(100\%\)

Multiply the decimal scale factor by \(100\) to write it as a percent.

Examples of Possible Scale Factors

New Side Length: \(1\)

\(\displaystyle k=\frac{1}{2}=0.5=50\%\)

The new square is one-half the original side length, so it is a reduction.

New Side Length: \(3\)

\(\displaystyle k=\frac{3}{2}=1.5=150\%\)

The new square is one and one-half times the original side length.

New Side Length: \(6\)

\(\displaystyle k=\frac{6}{2}=3=300\%\)

The new square has side lengths three times as large as the original.

Percent of the Original Size Versus Percent Increase

A scale factor of \(1.5\), or \(150\%\), means the new side length is \(150\%\) of the original side length. This represents a \(50\%\) increase, not a \(150\%\) increase.

Percent of Original

\(1.5=150\%\)

The new side length is \(150\%\) as large as the original side length.

Percent Increase

\(150\%-100\%=50\%\)

The amount above the original \(100\%\) is the percent increase.

🧠 Math Vocabulary

Scale factor: The multiplier used to enlarge or reduce a figure.
Original figure: The starting figure before a scale transformation is applied.
Scaled copy: A figure created by multiplying every length of an original figure by the same scale factor.
Similar figures: Figures with congruent corresponding angles and proportional corresponding side lengths.
Corresponding sides: Sides that occupy matching positions in two figures.
Ratio: A comparison of two quantities using division.
Equivalent forms: Different representations of the same number, such as \(\frac{3}{2}\), \(1.5\), and \(150\%\).
Fraction: A number written as a quotient of two integers.
Decimal: A base-ten representation of a number.
Percent: A ratio comparing a quantity to \(100\).
Enlargement: A scaled copy produced by a scale factor greater than \(1\).
Reduction: A scaled copy produced by a scale factor between \(0\) and \(1\).
Congruent: Having the same shape and the same size.
Dilation: A transformation that changes a figure’s size by a scale factor while preserving its shape.
Percent increase: The amount of increase expressed as a percent of the original value.
Coordinate plane: A two-dimensional system used to locate points using ordered pairs.

💡 Main Idea

The original square has a side length of \(2\) units. To determine the scale factor, divide the new side length by \(2\). For example, a new side length of \(3\) units produces \(k=\frac{3}{2}=1.5=150\%\). This means the new side length is \(150\%\) of the original, which represents a \(50\%\) increase.

📚 What You Should Already Know

Students should know how to divide numbers, simplify fractions, convert fractions to decimals, convert decimals to percents, locate figures on a coordinate plane, and recognize corresponding side lengths in similar figures.

🚀 What Comes Next

Students can extend this reasoning by drawing dilated figures on a coordinate plane, finding missing side lengths, comparing enlargement and reduction scale factors, studying how scale factor affects perimeter and area, and distinguishing percent of the original from percent increase or decrease.

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