Calculate a Proportion with Cross-Product Checks

Identify the missing quantity and its position

Suppose 8 notebooks cost 20 currency units at a constant unit price, and the cost of 14 notebooks is unknown. Write 8:20 = 14:X, keeping notebook counts first and costs second. The local calculator expects exactly that arrangement with the missing fourth value. Enter A = 8, B = 20 and C = 14. The resulting X = 35 has currency units. Merely listing the three numbers without deciding what they represent can produce a numerical result with the wrong meaning.

Derive and check the calculation

For A/B = C/X, cross multiplication gives AX = BC, and dividing by nonzero A gives X = BC/A. The notebook example yields 20×14/8 = 35. Verify it with 8×35 = 20×14 = 280. A unit-rate check gives 20/8 = 2.5 currency units per notebook and 14×2.5 = 35. These checks validate the arrangement as well as the arithmetic, provided the constant-price assumption is actually part of the problem.

Work a scale-drawing example

At a drawing scale of 1:50, a represented length of 7.2 centimetres corresponds to X centimetres in reality. Write 1:50 = 7.2:X to obtain X = 360 centimetres, or 3.6 metres. The final unit conversion is a separate step. Do not enter the drawing length in centimetres and then label the unconverted result as metres. The scale refers to corresponding lengths; an area represented by a scale drawing would use the square of the linear factor instead.

Rearrange another unknown position

For 6:X = 15:25, swap the two complete sides of the equality to obtain 15:25 = 6:X. This places the missing value fourth without changing the relationship. Enter A = 15, B = 25 and C = 6 to obtain X = 25×6/15 = 10. Swapping complete equal ratios preserves the equation; swapping only two entries within one ratio generally does not. Confirm the candidate in the original statement: 6/10 and 15/25 are both 0.6, and their cross-products both equal 150.

Interpret zero and rounding deliberately

The displayed formula cannot divide by A = 0. A zero in a proportion may imply no solution, multiple possibilities or an invalid quotient, depending on the other entries, so it requires separate analysis. For ordinary positive measurements, retain unrounded values while computing X and round only the final report. If the proportion gives 7.5 containers but containers must be bought whole, the purchasing rule may require eight. That practical rounding changes the purchased amount, not the exact proportional result, and should be stated explicitly. Record the unrounded result beside the operational quantity so another reader can reproduce the decision.

Keep units and the comparison order visible

A ratio compares the first named quantity with the second. The ratio of length to width differs from width to length, even though both describe the same rectangle. For quantities of the same kind, convert to matching units before simplifying: 1 metre to 25 centimetres is 100:25 = 4:1. For different kinds, the units form part of the rate, such as kilometres per hour or dollars per kilogram. Removing those labels can make a correct numerical quotient answer a different question. Write the two named quantities in the same order beside every corresponding ratio.

Check whether constant proportionality is justified

A proportional model assumes a constant multiplier between the quantities over the range being considered. Fixed fees, minimum charges, thresholds, waste and changing rates can violate that assumption. If one ticket costs 8 units plus a fixed booking fee of 3, a one-ticket total of 11 does not imply that four tickets cost 44; the stated fee rule gives 35 if the fee is charged once. The calculator solves the proportional equation entered, but cannot decide whether the real situation follows it. Confirm the relationship from the problem statement or the relevant pricing or measurement rule first.

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