Matemáticas
Ratio and Fraction: What Is the Difference?
With 12 boys and 18 girls the ratio is 2:3 and the fraction is 2/5. The same class, two different comparisons, and the conversion word problems assume you know.
Respuesta breve
What is the difference between a ratio and a fraction?
A ratio compares part with part; a fraction compares part with whole. With 12 boys and 18 girls the ratio of boys to girls is 2:3, but the fraction of the class that is boys is 2/5.
La respuesta corta
A ratio compares parts with each other; a fraction compares a part with the whole. A class of 12 boys and 18 girls has a boys-to-girls ratio of 2:3, while the fraction of the class that is boys is 2/5. Adding the parts of the ratio gives the denominator of the fraction.
Qué cambia la respuesta
- A ratio can compare more than two quantities - 2:3:5 is an ordinary ratio, and no single fraction says the same thing.
- Order matters in a ratio and not in a fraction: 2:3 and 3:2 describe opposite situations, while 2/5 has only one reading.
- A ratio can be written as a fraction and be perfectly correct, provided what you mean is part compared with part: 12:18 as a fraction is 2/3, meaning there are two thirds as many boys as girls.
- Whether the question hands you a part or the total decides which direction you convert in, and checking that first prevents most of the errors.
The same class, written both ways
Take a class of 30 with 12 boys and 18 girls. Every true statement about that class can be written as a ratio or as a fraction, and the two are not saying the same thing about it.
The ratio of boys to girls is 12:18, which simplifies to 2:3. Read that as: for every 2 boys there are 3 girls. Nothing in it mentions the size of the class, and it would be the identical ratio in a class of 10 with 4 boys and 6 girls, or in a school of 300.
The fraction of the class that is boys is 12/30, which simplifies to 2/5. Read that as: out of every 5 people in the room, 2 are boys. This one does refer to the whole, and it changes if the size of the group changes while the makeup does not.
- Boys to girls, part to part: 12:18 = 2:3
- Boys as a fraction of the class, part to whole: 12/30 = 2/5
- Girls as a fraction of the class, part to whole: 18/30 = 3/5
- Boys as a fraction of the girls, part to part written as a fraction: 12/18 = 2/3
Converting one into the other
There is one move in each direction, and word problems assume you can make it without being told to.
From ratio to fraction: add the parts to find how many shares there are in total, then put each part over that total. The ratio 2:3 has 2 + 3 = 5 shares, so boys are 2/5 of the class and girls are 3/5.
From fraction to ratio: the numerator is one part, and whatever is left of the denominator is the other. If 2/5 of the class are boys then 3/5 are girls, so the ratio of boys to girls is 2:3. The subtraction in the middle - 5 minus 2 - is the step people forget.
- Ratio 3:4 → 7 shares → 3/7 and 4/7
- Ratio 2:3:5 → 10 shares → 2/10, 3/10 and 5/10, or 1/5, 3/10 and 1/2
- Fraction 3/8 are red → 5/8 are not red → ratio of red to not red is 3:5
The reading that loses the marks
Nearly every mark lost on this topic comes from one substitution: reading the ratio of boys to girls is 2:3 as two thirds of the class are boys. It is not. Two thirds of 30 is 20, and the real number of boys is 12.
It happens because 2:3 looks like the fraction 2/3, and in one sense it is one - but the 3 underneath is the girls, not the class. The question to ask before writing anything down is: three what? If the number below the line is one of the parts, the comparison is part to part. If it is the total, it is a fraction of the whole.
Sharing questions make the same demand in reverse. Share PKR 6,000 in the ratio 2:3 means five shares of PKR 1,200 each, giving PKR 2,400 and PKR 3,600. The first quantity to find is always the value of one share, and the first mistake is always dividing by 2 or by 3 instead of by 5.
Which one a question actually wants
Ratios are the right instrument when the total is unknown or irrelevant: recipes scaled up and down, map scales, mixing concrete, exchange rates, gear teeth. Fractions are the right instrument when the whole is fixed and you want a share of it: probability, percentages, pie charts, discounts.
Probability is the clearest illustration of the two coexisting. A bag holds 3 red counters and 5 blue. The probability of drawing red is 3/8, a fraction of the whole bag. The odds of drawing red are 3:5, a ratio of red to not red. Both numbers are correct and they answer different questions, which is why a probability of 3/5 in that situation is a sign the ratio was used where the fraction was needed.
Percentages sit on the fraction side, since a percentage is a fraction with 100 fixed underneath. That is why a ratio always has to be converted before a percentage can be found. In a 2:3 ratio the first part is not 2 per cent and it is not 23 per cent - it is 2/5, which is 40 per cent.
Preguntas frecuentes
Can a ratio be written as a fraction?
Yes, if you mean part compared with part. The ratio 12:18 written as 12/18 simplifies to 2/3 and means there are two thirds as many boys as girls. What you cannot do is call that 2/3 the fraction of the class that is boys, which is 2/5.
How do you simplify a ratio?
Divide every part by their highest common factor. For 12:18 the HCF is 6, so both parts are divided by 6 to give 2:3. The process is the same as simplifying a fraction, with one extra requirement: it must be applied to every part, including the third one in a ratio like 6:9:15.
What does a ratio written as 1:n mean?
One unit of the first quantity for every n of the second. It is the standard form for map scales and for comparing two ratios quickly. To convert 4:10 to this form, divide both parts by 4, giving 1:2.5. Decimals are allowed here, which surprises students used to whole-number ratios.
If a group of 24 is split in the ratio 3:5, how many are in each part?
There are 3 + 5 = 8 shares, so one share is 24 ÷ 8 = 3 people. The two parts are 3 × 3 = 9 and 5 × 3 = 15. Check by adding: 9 + 15 = 24. That check takes two seconds and catches a division by the wrong number.
Does the order of a ratio matter?
Completely. The ratio of boys to girls being 2:3 and the ratio of girls to boys being 2:3 describe different classes. Write the labels above the numbers when the question mentions two named groups, because the mark is lost by reversing them far more often than by miscalculating.
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