Ratio, Proportion, Partnership, Mixtures, Alligation & Average

Ratio is a comparison statement, not a subtraction statement. If two values are in the ratio 12:18, the real structure is 2:3. This reduced form is what matters in CUET UG because every valid pair must be a common multiple of the reduced ratio.

When a total amount is divided in a given ratio, the workflow is simple: add the parts, find the value of one part, and then scale up. The same part logic works for marks, money, ages, ingredients, and student groups.

Proportion means equality of two ratios. If $a:b = c:d$, then $ad = bc$. Cross multiplication is the fastest tool in missing-term questions and saves a lot of time in entrance-exam arithmetic.

Direct proportion appears when one quantity rises with the other, such as cost and quantity at fixed price. Inverse proportion appears when one rises while the other falls, such as workers and days for a fixed task.

Partnership is ratio with a business context. If all partners invest for the same duration, profit share follows the capital ratio. If the time period differs, profit share follows capital multiplied by time.

If a partner joins late, leaves early, or changes capital midway, split the year into clear phases and compute capital-months for each phase. Once the products are ready, the rest is ordinary ratio simplification.

For example, Rs 50000 for 5 months and Rs 30000 for 7 months becomes $50000 \times 5 + 30000 \times 7$. This one discipline prevents most mistakes in partnership questions.

Alligation is a shortcut for mixture questions in which two components combine to form a mean value. If a cheaper item has value C, a dearer item has value D, and the mean is M, then the quantity ratio is obtained from opposite differences.

This works for tea-price, grain-price, alloy, concentration, milk-water, acid-solution, and similar questions. If the mean is closer to the cheaper value, the cheaper quantity must be larger.

Replacement questions use a different constant. When equal quantity is removed and replaced repeatedly, the fraction of original content left after one operation is $(V-x)/V$.

Average becomes easy once you move back to totals. Most CUET questions are solved faster by using total = average x number rather than average = total/number.

Missing-value questions, combined averages, and correction questions all depend on controlling the total. If two groups are combined, do not take the simple mean of the two averages unless the group sizes are equal.

Weighted average is especially important when group sizes differ. For example, if one class has 18 students and another has 22, their averages do not contribute equally to the combined average.

Error-correction questions are often one-line problems. If one observation was taken wrongly, the total changes by the same correction amount, and the average changes by correction divided by number of observations.

This module looks broad because it includes ratio, proportion, partnership, alligation, mixtures, replacement, and average. But the real engine is small: preserve the correct constant and write the right comparison.

In ratio questions, reduce immediately. In proportion, cross multiply. In partnership, compute capital-time products. In alligation, use opposite differences. In averages, go back to totals.

That one-layer classification makes the chapter much faster. Students usually lose time by treating every question as a fresh arithmetic puzzle instead of identifying its structure first.