Ratios

What does a ratio represent?

The relative sizes of two or more quantities.

How can the ratio of dogs to cats (6 dogs, 4 cats) be expressed?

• As numbers: 6 : 4
• As fractions: 6/10 : 4/10
• As decimals: 0.6 : 0.4
• As percentages: 60% : 40%

Where do ratios appear in everyday life?

Recipes, formulations, chemical mixtures, concentrations, and more.

What is the key rule when simplifying ratios?

You may divide or multiply each part by any non-zero number and the ratio stays equivalent.

Why simplify ratios?

To express them in smallest whole-number form.

Example: Simplify 6 : 4

Divide both by 2 → 3 : 2.

What must always be done with decimal, fractional, or percentage ratios?

Convert them to whole-number ratios.

Example: Simplify 0.1 : 0.12

Multiply both by 100 → 10 : 12 → 5 : 6.

Example: Simplify ½ : ¾

Bring to common denominator → 2/4 : 3/4 → 2 : 3.

Example: Simplify 25% : 150%

Convert to numbers → 25 : 150 → divide by 25 → 1 : 6.

If blue and yellow buttons are shown, what ratio might be asked for?

Ratio of blue : yellow.

What other ratio might be asked?

Ratio of red : blue.

What fraction question is asked?

“What fraction of the buttons are red?”

How to simplify 225 : 550?

Divide by 25 → 9 : 22.

How to simplify 45 minutes : 5.5 hours?

Convert hours → 5.5 h = 330 min → ratio = 45 : 330 → divide by 15 → 3 : 22.

How to simplify 0.63 : 0.9?

Multiply by 100 → 63 : 90 → divide by 9 → 7 : 10.

How to simplify 3½ : 2¼?

Convert to improper fractions → 7/2 : 9/4
Multiply both by 4 → 14 : 18 → divide by 2 → 7 : 9.

What are ratio pair problems?

Problems asking you to complete a ratio so it is equivalent to a reference ratio.

When two ratios are equal, what do we call them?

Proportional.

Example: In a cake recipe, 2 eggs : 400g flour.
If you have 3 eggs, how much flour?

2 : 400 = 3 : x → x = 600g.

Solve: 10 : 30 = 2 : x

Cross-multiply → 10x = 60 → x = 6.

Solve: x : 36 = 4 : 9

Cross-multiply → 9x = 144 → x = 16.

Solve: 40 : 1000 = x : 150

Cross-multiply → 1000x = 6000 → x = 6.

What does a concentration ratio express?

Amount of substance per volume (e.g., g per L).

Example: 120 g sugar in 1000 mL water.
How much sugar in 75 mL?

120 : 1000 = x : 75 → x = 9 g.

Example: Stock solution = 150 g/L.
To dissolve 12 g, how much water?

150 : 1000 = 12 : x → x = 80 mL.

150 g per 1000 mL; 12 g requires how many mL?

80 mL.

160 g per 1000 mL; how much sugar in 175 mL?

28 g.

500 g formulation contains 20 g active; how much in 40 g?

1.6 g.

What is the stepping-stone method used for?

Solving ratio pairs by scaling through convenient intermediate values.

Steps in the method?

• Write full ratio on top.

• Write incomplete ratio underneath.

• Find a “stepping stone” (simple divisor/multiplier).

• Apply the same steps to the unknown column.

• Arrive at the final missing value.

Example: 160 g : 1000 mL → value for 175 mL?

Work through stepping stones → 28 g.

Example: 20 g : 500 g → value for 40 g?

Stepping stones → 1.6 g.

How do you split a total amount into a ratio a : b : c?

• Add ratio parts → total parts.

• Divide full amount by total parts → value per part.

• Multiply per-part value by each ratio number.

Split 800 g in the ratio 2 : 5 : 3.

Total parts = 10 → 800/10 = 80 g per part →
2×80 = 160 g
5×80 = 400 g
3×80 = 240 g

Split 1180 g in ratio (1/4) : (2/5) : (1/3).

• Convert to whole numbers:
  LCM of 4,5,3 = 60 →
  1/4 = 15, 2/5 = 24, 1/3 = 20 → ratio = 15 : 24 : 20

• Total parts = 59

• 1180 / 59 = 20 g per part

• Multiply:
  15×20 = 300 g
  24×20 = 480 g
  20×20 = 400 g

Split 800 g in ratio 2 : 5 : 3.

160 g : 400 g : 240 g.

Split 1180 g in ratio (1/4) : (2/5) : (1/3).

300 g : 480 g : 400 g.

What are the four main skills in ratio calculations?

• Simplifying ratios to whole numbers.

• Completing proportional ratios.

• Using ratio pairs (stepping-stone method).

• Splitting quantities according to ratios.

What real-world fields use ratios constantly?

Chemistry, pharmacy, cooking, engineering, biology, and medicine.