Sharing in a ratio

A prize of ₹660, a bag of 90 marbles, a plot of land — real life rarely hands out totals in equal shares. Far more often a quantity has to be split fairly but unevenly, according to a stated ratio: two partners who invested different amounts, three friends who put in different amounts of work, a recipe scaled between two cooks. Sharing in a ratio is the arithmetic skill built for exactly this: taking a fixed total and a ratio like $4:7$ or $2:3:4$ , and finding each party's exact share. The method is not a new trick — it is the familiar unitary method wearing a different hat. Instead of finding the value of one unit (as in "cost of 1 pen"), you find the value of one part of the ratio, then scale it up for each party. Add the ratio terms to get the total number of parts, divide the total quantity by that number to get one part, then multiply by each person's share of parts. This article expands the core idea, its algebraic justification, and the ₹660 worked example already met in the shorter Pass-1 version, adding the three-way case, a stretch problem, and the traps that cost the most marks in this topic.

The core insight

To share a quantity $Q$ in the ratio $a:b$ (or $a:b:c$ ), divide $Q$ by the total number of parts $a+b$ (or $a+b+c$ ) to get the value of one part, then multiply that value by each party's own ratio term.

The core idea

Sharing in a ratio always follows the same three moves, whether two people or three (or more) are involved, and whatever the total quantity happens to be — money, marbles, land, or time.

1. Count the total parts. A ratio like $a:b$ does not describe "two shares" — it describes a total of $a+b$ equal parts, of which one party gets $a$ of them and the other gets $b$ . This is the step students most often skip, jumping straight to multiplying or dividing by $a$ or $b$ alone.

2. Find the value of one part. Once the total number of parts is known, dividing the whole quantity by that number gives the size of a single part — a fixed "unit" that every share is built from:

\text{value of 1 part} = \frac{Q}{a+b}

where: $Q$ is the total quantity being shared, and $a+b$ is the total number of equal parts in the ratio $a:b$ .

3. Scale up for each party. Each party's share is simply their own ratio number multiplied by the value of one part:

\text{first share} = a \times \frac{Q}{a+b}, \qquad \text{second share} = b \times \frac{Q}{a+b}

where: $a$ and $b$ are the two parties' ratio terms, and the two shares together must reconstruct the whole quantity $Q$ . The same three moves extend directly to three (or more) parties sharing in ratio $a:b:c$ : add all three terms to get $a+b+c$ total parts, divide $Q$ by that sum to get the value of one part, then multiply by each party's own term. Nothing about the method changes — only the number of terms being added.

Figure

A bar split into eleven equal parts, four parts shaded for Aakash and seven parts shaded for Bina, showing how a ratio four to seven divides a whole into total parts

Derivation AWhy the shares always add back up to the whole

The goal is to prove, using nothing beyond fraction algebra a Class 8 student already owns, that splitting a quantity $Q$ in ratio $a:b$ by the "value of one part" method always reconstructs the full quantity exactly — so the standard check (do the shares sum to the total) is not a lucky coincidence but a guaranteed feature of the method.

Step 1. Write the value of one part as $\dfrac{Q}{a+b}$ , following directly from dividing the total $Q$ by the total number of parts $a+b$ .

Step 2. Write the two shares in terms of this single value: the first party's share is $a \times \dfrac{Q}{a+b}$ and the second party's share is $b \times \dfrac{Q}{a+b}$ , since each party simply takes their own ratio number of parts.

Step 3. Add the two shares together. Because both terms share the same denominator $a+b$ , they combine directly:

a\left(\frac{Q}{a+b}\right) + b\left(\frac{Q}{a+b}\right) = \frac{aQ + bQ}{a+b} = \frac{(a+b)Q}{a+b}

Step 4. Cancel the common factor $(a+b)$ from numerator and denominator, leaving simply $Q$ . The two shares always add up to exactly the original quantity, for any values of $a$ , $b$ , and $Q$ — which is why "check that the shares sum to the total" is a completely reliable final step, not just good habit.

Worked example

Problem. ₹660 is to be shared between Aakash and Bina in the ratio $4:7$ . Find each person's share, and then extend the same bag-of-marbles style problem to three friends.

Step 1. Find the total number of parts: $4 + 7 = 11$ parts.

Step 2. Find the value of one part: $\dfrac{660}{11} = ₹60$ .

Step 3. Find each share by multiplying by the ratio terms: Aakash gets $4 \times 60 = ₹240$ , and Bina gets $7 \times 60 = ₹420$ .

Step 4. Check: $240 + 420 = 660$ , matching the original total exactly.

Answer: Aakash receives ₹240 and Bina receives ₹420.

Competitive stretch

A bag of 90 marbles is shared among Ravi, Sam and Tara in the ratio $2:3:4$ . Step 1. Total parts: $2+3+4 = 9$ . Step 2. Value of one part: $\dfrac{90}{9} = 10$ marbles. Step 3. Shares: Ravi gets $2 \times 10 = 20$ , Sam gets $3 \times 10 = 30$ , Tara gets $4 \times 10 = 40$ marbles, and $20+30+40 = 90$ checks out. Now reverse it: if a problem only tells you Ravi received 20 marbles from a bag shared in ratio $2:3:4$ , you can still find the value of one part directly from Ravi's own share, $20 \div 2 = 10$ marbles per part, and then recover the whole bag size as $10 \times 9 = 90$ — the method works forwards (total to shares) and backwards (one share to total) with equal ease.

Common traps

Dividing by the number of people instead of the number of parts. In a $4:7$ split there are 2 people but 11 parts; dividing 660 by 2 answers a different, equal-sharing question, not this one.
Skipping the "add the ratio terms" step. Jumping straight to $660 \times 4$ or $660 \div 4$ without first finding $a+b$ produces a number with no meaning in this problem.
Mismatching names and numbers. If the ratio is stated "Aakash : Bina = 4 : 7," swapping the order partway through (giving Aakash the larger share) is a common slip — always keep names and ratio terms in the same order they were given.
Skipping the final check. Verifying that all shares add back up to the original total catches almost every arithmetic error and should always be the last line of working.

Ask the tutor: Not sure how many parts a ratio like $3:5:2$ makes, or want another sharing problem worked through step by step? Ask the AI tutor.

Why do we add the ratio terms first instead of just using them straight away?

Because the ratio terms only tell you how many parts each party gets, not the size of a part. A ratio $4:7$ means the whole is split into $4+7=11$ equal parts, four going to one side and seven to the other. Without adding the terms first, you have no way to know what fraction of the total quantity a single "part" represents, so any multiplication or division done before that step produces a meaningless number.

Does the method change if there are three or four people sharing instead of two?

No — the same three moves apply exactly as before. Add all the ratio terms to get the total number of parts (for example $2+3+4=9$ for three people), divide the total quantity by that sum to get the value of one part, then multiply by each person's own ratio term. The only difference from a two-way split is that there are more terms to add and more shares to compute at the end; the underlying logic never changes.

If I only know one person's share and the ratio, can I still find the total?

Yes. Since one part equals that person's share divided by their own ratio term, you can find the value of one part directly from any single share, without ever being told the overall total. Once you have the value of one part, multiply it by the total number of parts (the sum of all the ratio terms) to recover the full quantity, or by any other person's ratio term to find their share instead. This "working backwards" version of the method is common in exam questions that deliberately withhold the total.

Sharing in a ratio

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads