Simplifying and Equivalent Ratios

Both ratios simplify to 2 : 3. For 4 : 6, divide both parts by 2 to get 2 : 3. For 10 : 15, divide both parts by 5 to get 2 : 3. Since both ratios have the same simplest form, they are equivalent.

Visually, they both represent the same proportion:

The “split point” is in exactly the same place for both.

To handle decimal ratios, multiply both parts by 10 to eliminate the decimals: 0.6 × 10 = 6 and 0.9 × 10 = 9, giving 6 : 9. Then simplify by dividing both parts by 3: 6 ÷ 3 = 2 and 9 ÷ 3 = 3, giving 2 : 3.

You can verify: in the ratio 0.6 : 0.9, the first part is two-thirds of the second (0.6 ÷ 0.9 = \( \frac{2}{3} \)), and in 2 : 3, the first part is also two-thirds of the second (2 ÷ 3 = \( \frac{2}{3} \)). The relationship between the parts is identical, confirming equivalence.

In 3 : 5, for every 3 of the first quantity there are 5 of the second — the first quantity is smaller than the second. In 5 : 3, for every 5 of the first quantity there are 3 of the second — the first quantity is now larger. These describe opposite situations. For example, if a recipe uses flour and sugar in the ratio 3 : 5, you need more sugar than flour. If the ratio were 5 : 3, you’d need more flour than sugar. This is the “order doesn’t matter” misconception — students sometimes treat a : b and b : a as interchangeable.

You can also check numerically: 3 ÷ 5 = 0.6, but 5 ÷ 3 ≈ 1.67. Since 0.6 ≠ 1.67, the ratios are not equivalent. Order matters in ratios — swapping the parts creates a different ratio unless both parts are equal.

To eliminate the fraction, multiply both parts by 2: \( 1\frac{1}{2} \times 2 = 3 \) and \( 2 \times 2 = 4 \), giving 3 : 4. This works because multiplying both parts of a ratio by the same number produces an equivalent ratio.

Alternatively, convert to an improper fraction first: \( 1\frac{1}{2} = \frac{3}{2} \). So the ratio is \( \frac{3}{2} : 2 \), which is the same as \( \frac{3}{2} : \frac{4}{2} \). Since both parts have the same denominator, we can compare the numerators: 3 : 4. When a ratio contains fractions or mixed numbers, the key strategy is to multiply both parts by the lowest common denominator of the fractional parts to obtain whole numbers.

Example: 6 : 8 (multiply both parts by 2)

Another: 9 : 12 (multiply both parts by 3)

Creative: 1 : \( 1.\dot{3} \) or 1 : \( \frac{4}{3} \) (a Unitary Ratio). This is useful for comparison, showing that for every 1 of the first part, there is \( 1\frac{1}{3} \) of the second.

Trap: 4 : 5 — a student might add 1 to each part of 3 : 4, thinking this gives an equivalent ratio. But 4 : 5 ≠ 3 : 4 (since \( \frac{4}{5} = 0.8 \) while \( \frac{3}{4} = 0.75 \)). This is the additive misconception — equivalent ratios are formed by multiplying both parts by the same number, not by adding.

Example: 12 : 18 (12 = 6 × 2, 18 = 6 × 3, and HCF(2, 3) = 1, so HCF(12, 18) = 6)

Another: 6 : 30 (6 = 6 × 1, 30 = 6 × 5, and HCF(1, 5) = 1, so HCF(6, 30) = 6)

Creative: 42 : 54 (42 = 6 × 7, 54 = 6 × 9, and HCF(7, 9) = 1, so HCF(42, 54) = 6 — students rarely think of these less obvious multiples of 6)

Trap: 12 : 24 — a student might think “both are divisible by 6, so the HCF must be 6.” But HCF(12, 24) = 12, not 6. The trap exploits the common-factor-vs-HCF confusion — just because 6 is a common factor doesn’t make it the highest. You must check whether any larger number also divides both parts.

Example: 3 : 7 (HCF(3, 7) = 1, so both parts share no common factor other than 1)

Another: 4 : 9 (HCF(4, 9) = 1)

Creative: 13 : 17 (both are prime numbers, so HCF = 1 — students often assume larger numbers can always be simplified further)

Trap: 9 : 15 — a student might think “9 and 15 don’t share an obvious common factor” because neither is even. But both are divisible by 3: 9 ÷ 3 = 3 and 15 ÷ 3 = 5, so it simplifies to 3 : 5. The “no obvious factor” misconception catches students who only check for a factor of 2 and miss other common factors like 3 or 5.

Example: 12 : 18 and 20 : 30 (both simplify to 2 : 3)

Another: 15 : 25 and 21 : 35 (both simplify to 3 : 5)

Creative: 14 : 49 and 22 : 77 (both simplify to 2 : 7 — 14 ÷ 7 = 2, 49 ÷ 7 = 7; 22 ÷ 11 = 2, 77 ÷ 11 = 7)

Trap: 10 : 15 and 20 : 25 — a student might add 10 to each part to get the second ratio. But 10 : 15 simplifies to 2 : 3, while 20 : 25 simplifies to 4 : 5. Since 2 : 3 ≠ 4 : 5, these are not equivalent. This is the additive misconception again — adding the same number to both parts does not preserve the ratio.

This is a very common misconception. It is sometimes true, but only in a trivial case. When both parts are already equal (e.g. 3 : 3), adding the same number gives another equal ratio (e.g. 5 : 5), which is still equivalent to 1 : 1. But when the parts differ, it fails: 2 : 5 with 3 added to each gives 5 : 8, and 2 : 5 ≠ 5 : 8 (since \( \frac{2}{5} = 0.4 \) but \( \frac{5}{8} = 0.625 \)).

Equivalent ratios are created by multiplying both parts by the same number, not by adding. This is the additive misconception — one of the most common errors students make when working with ratios.

This is always true. If both parts of a ratio are even, they are both divisible by 2, meaning they share a common factor of 2. A ratio in its simplest form has no common factors other than 1, so any ratio with two even numbers can definitely be simplified further. For example, 8 : 14 can be divided by 2 to give 4 : 7.

Students might think they need to find the HCF to simplify, but recognising that “both even → not simplest form” is a useful shortcut. Note: this does not mean the ratio simplifies by dividing by exactly 2 — the HCF might be larger (e.g. 12 : 8 has HCF 4, not 2).

This is never valid. To keep a ratio equivalent, you must divide (or multiply) both parts by the same number. If you divide the first part by one number and the second part by a different number, you change the relationship between the parts.

For example, starting with 12 : 18 and dividing the first part by 3 and the second by 2 gives 4 : 9, but 12 : 18 = 2 : 3, and 4 : 9 ≠ 2 : 3. This is the “dividing by different numbers” error — the rule is simple: whatever you do to one part, you must do exactly the same to the other.

This is sometimes true, but only for ratios like 1 : 1 where the difference is 0. For most ratios, the difference scales as the ratio scales.

Consider 2 : 5 (difference is 3). If we multiply by 2 to get 4 : 10, the difference becomes 6. If we multiply by 10 to get 20 : 50, the difference becomes 30. The difference multiplies by the same factor as the ratio itself. Focusing on the “gap” rather than the multiplier is a symptom of additive thinking.

(All three ratios are equivalent to 2 : 3)

The student has found an equivalent ratio but has not fully simplified. This is the “partial simplification” misconception — thinking that dividing by any common factor is enough. While 6 : 4 is equivalent to 12 : 8, both 6 and 4 still share a common factor of 2. The HCF of 12 and 8 is 4, so the fully simplified form is 12 ÷ 4 : 8 ÷ 4 = 3 : 2.

To fully simplify a ratio, students should divide both parts by the highest common factor, or keep dividing until no common factors remain. From 6 : 4, dividing by 2 again gives 3 : 2, which is the simplest form since HCF(3, 2) = 1.

The student has applied additive thinking to a multiplicative concept. Simplification requires division, not subtraction. Subtraction changes the ratio entirely.

Correct approach: Find the highest common factor (HCF) of 24 and 36, which is 12. Then divide both parts by 12.
\(24 \div 12 = 2\)
\(36 \div 12 = 3\)
So the simplified ratio is 2 : 3.

The student has applied the additive misconception. 15 : 18 simplifies to 5 : 6, which is not the same as 5 : 8. Think about mixing paint:

The correct approach is multiplicative: 15 is 3 times 5, so we must multiply 8 by 3 to get 24. The equivalent ratio is 15 : 24.

The student has made the “different units” error. You cannot compare 40 of one thing to 1 of another directly.

To write this ratio correctly, convert both quantities to the same unit. Since 1 hour = 60 minutes, the ratio becomes 40 : 60.

Now simplify as a fraction: \( \frac{40}{60} = \frac{4}{6} = \frac{2}{3} \).
The simplest form is 2 : 3.