Match the shared quantity before scaling.
Review ratio parts, chained comparisons, scaling and unequal denominators with fresh calculations. Six-item counts are local practice, not a certified score.
Each of six original items contributes one correct answer. The count describes this edition’s fictional quantities and assumptions, not a speed, ability or job-readiness band. All six worked answers and this full guide remain free at every count.
A ratio compares parts, not automatically the whole
A ratio such as 2:7 compares two quantities in that order. When those are the only two groups, the whole has nine equal parts. When one group’s amount is supplied instead, use its corresponding part count to find the scale. These six items distinguish those starting points. Write the labels next to the ratio before calculating; reversing water and concentrate or blue and red can produce a perfectly consistent answer to the wrong question.
Work a new sharing example
With a 2:7 split of 81 fictional tokens, one part is 81 ÷ 9 = 9. The two groups receive 18 and 63, which add back to 81 and simplify to 2:7. Check both the total and the ratio: meeting only one is insufficient. If 81 described the larger group alone instead of the combined total, this calculation would not apply. The wording determines which quantity is available, so do not reuse a formula without its condition.
Unify the middle term in a chain
If A:B = 3:4 and B:C = 5:6, give both representations of B the same value. Scaling the first by five gives 15:20 and the second by four gives 20:24. Therefore A:C = 15:24 = 5:8. Directly joining 3 and 6 treats two different sizes of B as if they matched. Positive quantities are assumed here so the ordinary ratio divisions are defined. A shared name alone does not establish a shared numeric scale.
Direct proportion needs a fixed per-unit relationship
An original recipe example uses 640 grams for eight equal servings. At the same grams per serving, five servings need 640 ÷ 8 × 5 = 400 grams. This is justified by the explicit constant-per-serving assumption. A fixed setup amount or a non-additive process would need a different model. Keep the mass and serving units visible, and check that a smaller number of identical servings requires less of the ingredient under this particular model.
Compare rates after aligning denominators
For two fictional groups, 18 completed out of 30 is 60%, while 42 out of 60 is 70%. The second share is higher by ten percentage points. Subtracting completed counts gives 24, a different unit and question. You can compare fractions by a common denominator, cross multiplication or decimals, provided the denominators are valid. Choose the representation that makes the requested difference clear rather than rounding each group early and amplifying a rounding error.
Use a ratio check in reverse
After scaling, divide the computed quantities by their original ratio parts. They should recover the same one-part scale. For a chained ratio, substitute your answer into both original relationships rather than checking only the final simplified pair. If the output has a specified rounding rule, preserve exact fractions until the last step. The worked explanations show every intermediate quantity needed here. A calculator is allowed, but it cannot tell whether you attached the amount to the correct named group.
Keep proportional practice separate from prediction
A count out of six records answers about the stated fictional shares, scales and conditions. It supplies no population norm, intelligence estimate, hiring threshold or evidence of performance under pressure. The tool does not claim that every real recipe, production process or group comparison is proportional. Repeating familiar values after reading the keys may reward recall. To practise transfer, create a fresh labeled example, justify the denominator or scaling assumption, and verify both the result and its units independently.