How to Do Scaling by Fractions and Mixed Numbers?

Scaling multiplies a quantity by a fraction or mixed number, and the result tells you the direction at a glance: multiply by less than 1 and the quantity shrinks, by more than 1 and it grows, by exactly 1 and it stays put. Below: the method and worked examples.

Scaling is an essential mathematical concept that helps us to adjust the size of a given quantity while maintaining its proportionality.

Whether you are a beginner or an advanced learner, this article will provide you with a clear and concise guide to scaling by fractions and mixed numbers.

A step-by-step guide to scaling by fractions and mixed numbers

There are some scaling rules:

• Whenever you multiply a fraction by a number smaller than \(1\) whole, its size gets scaled down.
• Whenever you multiply a fraction by a number larger than \(1\) whole, its size gets scaled up.

Scaling by fractions and mixed numbers can be done in just a few simple steps. Here is a step-by-step guide to help you master this skill:

Step 1: Understand the Problem: Read the problem carefully to determine what needs to be scaled and by what amount. This will help you choose the appropriate fraction or mixed number to use in the scaling process.

Step 2: Write the Original Quantity: Write down the original quantity that needs to be scaled. This will help you keep track of the starting point as you perform the scaling operation.

Step 3: Write the Fraction or Mixed Number: Write down the fraction or mixed number that you will use to scale the original quantity. This will be the amount by which you want to increase or decrease the original quantity.

Step 4: Convert Mixed Numbers to Improper Fractions If you are using a mixed number, convert it to an improper fraction. To do this, multiply the whole number by the denominator of the fraction, and add the numerator. Write the result over the denominator to get the improper fraction.

Step 5: Multiply or Divide To scale by a fraction, multiply the original quantity by the fraction. To scale by a mixed number, multiply the original quantity by the mixed number’s improper fraction. If you are scaling down, multiply the original quantity by a fraction less than 1.

Step 6: Simplify the Result (if necessary): If the result is an improper fraction, simplify it by reducing it to its lowest terms. If the result is a mixed number, convert it back to a mixed number if necessary.

Step 7: Write the Final Answer: Write down the final answer, including the units of measurement if applicable. Make sure to label the answer appropriately to avoid confusion.

Here’s an example to help illustrate these steps:

Example: You have a recipe that calls for \(\frac{3}{4}\) cups of flour, but you need to make a double batch. How much flour will you need?

Step 1: Understand the Problem: We need to scale the original quantity of flour by a factor of \(2\).

Step 2: Write the Original Quantity: The original quantity is \(\frac{3}{4}\) cup of flour.

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Step 3: Write the Fraction: We need to scale by a factor of \(2\), so we will use the fraction \(\frac{2}{1}\).

Step 4: Convert Mixed Numbers to Improper Fractions Not applicable in this example.

Step 5: Multiply or Divide To scale by the fraction, we multiply the original quantity by the fraction: \(\frac{3}{4} \times \frac{2}{1} = \frac{6}{4}\)

Step 6: Simplify the Result: The result is an improper fraction, so we simplify it to its lowest terms: \(\frac{6}{4} = \frac{3}{2}\)

Step 7: Write the Final Answer: The final answer is \(\frac{3}{2}\) cup of flour. We can also express this as \(1\frac{1}{2}\) cups of flour.

Scaling by Fractions and Mixed Numbers, Example 1

Which expression is greater or least? \(2×5 \frac{2}{3} …. 11\)

Solution:

Step 1: Calculate: (2×5 frac{2}{3}=2×frac{17}{3}=\frac{34}{3}=11 \frac{1}{3}).
Step 2: Compare them. \(10 \frac{4}{3}\) is less than \(11\). So, \(2×5 \frac{2}{3}<11\)

Scaling by Fractions and Mixed Numbers, Example 2

Which expression is greater or least? \(3×4 \frac{3}{4} …. 13\)

Solution:

Step 1: Calculate: \(3×4 \frac{3}{4}=3×\frac{19}{4}=\frac{57}{4}=14 \frac{1}{4}\)
Step 2: Compare them. \(14 \frac{1}{4}\) is greater than \(13\). So, \(3×4 \frac{3}{4}>13\)

Exercises for Scaling by Fractions and Mixed Numbers

Compare each expression.

1. \(\color{blue}{6\times 3\frac{10}{7}………15}\)
2. \(\color{blue}{4\times 2\frac{3}{7}……… 12}\)
3. \(\color{blue}{3\times 4\frac{5}{9}……… 13}\)

1. \(\color{blue}{6\times 3\frac{10}{7}\:>15}\)
2. \(\color{blue}{4\times 2\frac{3}{7}<12}\)
3. \(\color{blue}{3\times 4\frac{5}{9}>13}\)

Predict the answer before you multiply

You can tell whether a scaled amount will grow or shrink just by comparing the scale factor to 1. This is a quick way to check your work and to answer comparison questions without doing the full calculation.

For a question like 5 × 7/8 … 5, you do not need to multiply. Since 7/8 is less than 1, the product must be less than 5.

Worked example: scaling up and down with mixed numbers

Scaling up: A recipe uses 2/3 cup of sugar. You want to make 1½ batches.

1. Change the mixed number to an improper fraction: 1½ = 3/2.
2. Multiply: 2/3 × 3/2 = 6/6.
3. Simplify: 6/6 = 1. You need 1 cup of sugar.

Scaling down: A board is 8 feet long and you need a piece that is 3/4 as long.

1. Multiply by the fraction: 8 × 3/4 = 24/4.
2. Simplify: 24/4 = 6. The piece is 6 feet long.

Notice that both times you multiply. Scaling down does not mean dividing by the fraction. It means multiplying by a fraction less than 1. To undo a scaling, multiply by the reciprocal: 6 × 4/3 = 8 brings the board back to its full length.

Frequently Asked Questions

What does scaling fractions mean?

Scaling with fractions means resizing a number by multiplying it by a fraction or mixed number. The fraction is called the scale factor. A factor such as 1/2 makes the amount half as big, and a factor such as 2½ makes it two and a half times as big. You always multiply, then simplify the answer.

Does multiplying by a fraction always make a number smaller?

No. Only fractions less than 1, like 3/4 or 2/5, make a positive number smaller. A fraction equal to 1, like 5/5, leaves it unchanged. Improper fractions and mixed numbers greater than 1, like 7/4 or 1⅓, make it larger. For example, 9 × 1⅓ = 12.

How do you scale a recipe by a mixed number?

Write the mixed number as an improper fraction, then multiply each ingredient by it. To make 2½ batches of a recipe that uses 3/4 cup of milk, compute 3/4 × 5/2 = 15/8, which is 1⅞ cups. Simplify each answer and turn improper fractions back into mixed numbers so they are easy to measure.

What is a scale factor with fractions?

A scale factor is the number you multiply by to resize something, and it can be a fraction. In scale drawings, a factor of 1/4 means every length in the copy is one fourth of the original. If a side is 12 cm, the scaled side is 12 × 1/4 = 3 cm.

How do you undo scaling by a fraction?

Multiply by the reciprocal of the scale factor. If a length was scaled by 2/3, multiply the new length by 3/2 to get the original back. For example, 9 × 2/3 = 6, and then 6 × 3/2 = 9. The two factors cancel because 2/3 × 3/2 = 1.