Writing Ratios

A ratio describes a relationship in a specific order. If there are 3 boys and 5 girls, the ratio of boys to girls is 3 : 5 (boys are the smaller group), but 5 : 3 implies boys are the larger group.

Visually, these are different distributions. Mixing paint in a ratio of 3 parts red to 5 parts blue creates a different colour than 5 parts red to 3 parts blue.

Both ratios describe the same proportional relationship. In 4 : 10, for every 4 of the first quantity there are 10 of the second. Dividing both parts by 2 gives 2 : 5.

Visually, the total length occupied by the parts remains consistent relative to each other. You can group the 4 and 10 into two groups of (2 and 5).

A ratio tells us the relative amounts, not the actual amounts. 1 : 3 means that for every £1 one person gets, the other gets £3 — but the “£1” acts as a unit that can represent any amount.

If they share £800 in the ratio 1 : 3, there are 4 parts total. Each part is worth £200. So one person gets £200 and the other gets £600. Both are greater than £100.

Ratios can compare three or more quantities simultaneously. 2 : 3 : 5 simply means that for every 2 of the first item, there are 3 of the second and 5 of the third. This is very common in recipes (e.g., flour, sugar, butter) or manufacturing (e.g., sand, cement, aggregate). The math works exactly the same way: the total number of parts is 2 + 3 + 5 = 10.

This addresses the Part-to-Whole misconception. 1 : 4 compares the parts against each other (Part-to-Part). To find the fraction of the total, you must add the parts together first.

There is 1 Red part and 4 Blue parts, making 5 parts in total. Therefore, Red is 1 out of 5, or \(\frac{1}{5}\) of the total, not \(\frac{1}{4}\). The fraction \(\frac{1}{4}\) would mean the ratio was 1 : 3.

Example: 6 : 8 (dividing both by 2 gives 3 : 4)

Another: 15 : 20 (dividing both by 5 gives 3 : 4)

Creative: 75 : 100 — both numbers are greater than 50, and dividing both by 25 gives 3 : 4.

Trap: 4 : 5. A student might think “I just added 1 to both sides” (3+1 : 4+1). Ratios are multiplicative, not additive. 4:5 is not equivalent to 3:4.

Example: 4 boys and 6 girls (4 : 6 simplifies to 2 : 3)

Another: 10 boys and 15 girls (10 : 15 simplifies to 2 : 3)

Creative: 14 boys and 21 girls — that’s 35 children altogether, and 14 : 21 simplifies to 2 : 3.

Trap: “There are 2 boys and 3 girls in Class A, and 2 boys and 3 girls in Class B, so the ratio of boys in Class A to boys in Class B is 2:3.” This is confusing the ratio within groups with the ratio between groups. The ratio of boys in A to boys in B is actually 1:1.

Example: 30 cm and 2 m. Converting to the same unit: 30 cm and 200 cm, giving 30 : 200 = 3 : 20.

Another: 500 m and 2 km. Converting: 500 m and 2000 m, giving 500 : 2000 = 1 : 4.

Creative: 4 m and 600 mm. Converting to mm: 4000 mm and 600 mm, giving 4000 : 600 = 20 : 3.

Trap: 50 cm to 3 m. A student writes 50 : 3 without converting, getting a completely wrong ratio. The correct approach is to convert to the same unit: 50 cm and 300 cm gives 50 : 300 = 1 : 6.

The total is \( a + b \). Half the total is \( \frac{a + b}{2} \). We need \( a < \frac{a + b}{2} \), which simplifies to \( a < b \). So this is true when \( a < b \), but false when \( a \geq b \).

True case: 2 : 5 → \( a = 2 \), total = 7, half = 3.5, and 2 < 3.5. False case: 5 : 2 → \( a = 5 \), total = 7, half = 3.5, and 5 > 3.5. In 1 : 1, \( a = 1 \) equals half of 2, so it is also false.

Simplifying a ratio means dividing both parts by the same number greater than 1. This is only possible if both parts share a common factor greater than 1. If both parts are coprime (HCF = 1), the ratio is already in its simplest form and cannot be simplified further.

A ratio can be written in any order depending on what is being compared. If there are 8 cats and 3 dogs, the ratio of cats to dogs is 8 : 3 (first number larger), but the ratio of dogs to cats is 3 : 8 (first number smaller).

Ratios express a multiplicative relationship between like quantities. You cannot compare length to mass directly. Even if the units are both lengths (cm and m), the numbers cannot be put into a ratio until they represent the same unit.

The student has reversed the order of the ratio. This is the “order doesn’t matter” misconception. A ratio must follow the order stated in the question: red to blue means red first, then blue.

The correct answer is 7 : 3.

The student is using subtraction (finding the difference) instead of multiplicative comparison. This is the “ratio as difference” misconception.

There are 8 strawberry and 4 lemon. The ratio is 8 : 4, which simplifies to 2 : 1.

The student is treating ratio simplification as an additive process (subtraction) rather than a multiplicative process (division). Ratios are only preserved by multiplying or dividing both parts by the same number.

If you have a photo that is 6cm by 9cm and you cut 2cm off each side, the shape changes (it becomes less rectangular). To keep the ratio, you must divide. The HCF of 6 and 9 is 3. \( 6 \div 3 = 2 \) and \( 9 \div 3 = 3 \). The correct answer is 2 : 3.

The student forgot to convert to the same units first (the “ignoring units” misconception). 2 m = 200 cm.

The correct ratio is 40 : 200, which simplifies to 1 : 5. The student’s answer (20 : 1) implies the first length is 20 times bigger than the second, whereas in reality, 2 metres is 5 times bigger than 40cm.