5.2: Applying ratios

A ratio expresses a relationship between two quantities in a mixture or combination. For example, an orange paint might consist of 3 parts red paint for every 4 parts yellow paint, written “3 : 4” (3 to 4). Ratios stay the same if we multiply each quantity by the same number. For example, 3 : 4 is equivalent to 6 : 8, 9 : 12, 12 : 16, etc.

Apply a ratio to one quantity

Here are a couple of examples of ratio problems where we have to find the amount of one quantity to mix with a specified quantity to obtain a particular ratio.

• Word problem: Frida has 6 gallons of red paint. How much yellow paint should Frida mix it with to make orange paint that is 3 parts red paint for every 4 parts yellow paint?
• Ratio problem: Create an equivalent ratio for 3 : 4 expressed as 6 : something.
• Multiply by 6/3: 3 x 6/3 = 6 gallons of red paint.
• Frida needs 4 x 6/3 = 4 x 2 = 8 gallons of yellow paint.
• Check: equivalent ratios are 3 : 4, 3×2 : 4×2, 6 : 8.

• Word problem: Irma’s favourite margarita recipe uses 1 1/2 (one and a half) ounces of tequila and 1 ounce of triple sec. How much triple sec should Irma mix with 25 ounces of tequila to make the perfect margarita mix for her party?
• Ratio problem: Create an equivalent ratio for 1 1/2 : 1 expressed as 25 : something.
• 1 1/2 : 1 is equivalent to 3 : 2.
• Multiply by 25/3: 3 x 25/3 = 25 ounces of tequila.
• Frida needs 2 x 25/3 = 50/3 = 16 2/3 (sixteen and two-thirds) ounces of triple sec.
• Check: equivalent ratios are 3 : 2, 3×25 : 2×25, 75 : 50, 75/3 : 50/3, 25 : 50/3.

Apply a ratio to the total

Here are a couple of examples of ratio problems where we have to find the amount of each quantity to mix in a particular ratio to obtain a specified total.

• Word problem: Raymond needs 14 gallons of orange paint that is 3 parts red paint for every 4 parts yellow paint. How many gallons of each colour paint does Raymond need?
• Ratio problem: Apply the ratio 3 : 4 to the total 14.
• Add the parts: 3 + 4 = 7.
• Divide the total (14) by 7 parts to get 14/7 = 2 gallons of paint in each part.
• Raymond needs 3 x 2 = 6 gallons of red paint.
• Raymond needs 4 x 2 = 8 gallons of yellow paint.
• Check: equivalent ratios are 3 : 4, 3×2 : 4×2, 6 : 8.
• Check: 6 gallons red + 8 gallons yellow = 14 gallons orange.

• Word problem: Ivan wants to make 30 ounces of margarita mix for his party based on Irma’s ratio of 1 1/2 (one and a half) ounces of tequila for every ounce of triple sec. How many ounces of each liquor does Ivan need?
• Ratio problem: Apply the ratio 1 1/2 : 1 to the total 30.
• 1 1/2 : 1 is equivalent to 3 : 2.
• Add the parts: 3 + 2 = 5.
• Divide the total (30) by 5 parts to get 6 ounces of liquor in each part.
• Ivan needs 3×6 = 18 ounces of tequila.
• Ivan needs 2×6 = 12 ounces of triple sec.
• Check: equivalent ratios are 3 : 2, 3×6 : 2×6, 18 : 12.
• Check: 18 ounces of tequila + 12 ounces of triple sec = 30 ounces of margarita mix.

The video below works through some examples of applying ratios.