Chapter Summary and Practice

After a gap, ratio questions can look similar even when they ask for different things. One asks you to resize a picture, another to divide a total, and another to compare prices for equal amounts. This revision page rebuilds the ideas in the order you learned them. Use it to recover the meaning behind each method, not just the arithmetic steps. A reliable solution begins by identifying the quantities and the relationship between them.

Read each reminder, pause at the worked examples and choose a method before looking at the solution. Finish with the mixed questions, then use the Binairo activity to practise careful logical reasoning.

7.1 Observing Similarity in Change

From Lesson 1: proportional change multiplies corresponding quantities by one common factor. When the same image is resized without cropping, width and height must use that same factor to preserve its proportions. A factor above 1 enlarges it, while a positive factor below 1 reduces it.

The sizes 60 by 40, 30 by 20 and 90 by 60 share the same proportions. Subtracting 20 from both dimensions of 60 by 40 instead gives 40 by 20. The factors 2/3 and 1/2 differ, so equal subtraction is not the same as proportional scaling.

The practical method is: divide new by old for one corresponding quantity, then multiply the other old quantity by that factor. Being rectangular is not by itself evidence that an image has been resized proportionally.

7.2 Ratios

From Lesson 2: a ratio a : b compares two quantities in a stated order. For every a units of the first, there are b units of the second. The numbers are called terms. Reversing the comparison reverses the terms.

Equivalent ratios are produced by multiplying or dividing both terms by the same positive factor. The actual amounts can differ: 3 : 2, 6 : 4 and 9 : 6 express the same relationship. A part-to-part ratio is not automatically a part-to-whole comparison.

7.3 Ratios in their Simplest Form

For whole-number terms, divide both by their HCF. Thus 72 : 96 simplifies by 24 to 3 : 4. If the remaining terms still share a factor larger than 1, the ratio is not yet simplest.

Two ratios are in proportion when they are equivalent; their simplest forms match. Write a : b :: c : d. Keep quantity order consistent: width : height must be compared with width : height, not height : width.

7.4 Problem Solving with Proportional Reasoning

From Lesson 3: use ratios to compare quantities fairly. A lemonade recipe keeps the same sweetness when both its glass count and sugar amount scale together, assuming equal glass and spoon sizes. A coffee mixture’s strength depends on decoction relative to milk, not the amount of decoction alone.

Regular coffee has decoction : milk = 3 : 7. A mixture 24 : 56 is equivalent. Ratios 1 : 2 and 3 : 10 are respectively stronger and lighter than regular coffee. Compare at a common milk amount or common total before judging.

For missing terms, use the factor attached to a known pair. From 14 : 21 to 6 : x, the factor is 3/7, so x = 9. Fractional factors and fractional ratio terms are allowed.

Equal additions generally change a ratio. Neelima and her mother change from ages 3 and 30 to 12 and 39: the ratio changes from 1 : 10 to 4 : 13, while the difference remains 27 years. This distinguishes an additive relationship from a proportional one.

For visual activities, compare corresponding dimensions and use one scale factor for every measured length. In a repeating brick pattern, count one complete repeat unit: repeating it multiplies both colour counts equally. Different-sized drawings can all be valid when their corresponding ratios agree.

Trairasika — The Rule of Three

From Lesson 4: write c = fa and d = fb for one common factor f. Then c/a = d/b. Multiplying by ab produces ad = bc. For the positive quantities in these problems, this cross-product condition tests a proportion as well as helping solve it.

If d is unknown, divide by a to obtain d = bc/a. The traditional names are pramāṇa for a, phala for b, ichchhā for c and ichchhāphala for d. The structure means “given result times requested measure, divided by given measure.”

Before calculating, align corresponding units. Time in minutes must match time in minutes; a weight in grams must match the other weight in grams. Two different quantities, such as time and distance, may naturally have different units from each other.

Check that direct proportion applies. At constant speed, more time gives proportionally more distance. For a fixed journey, higher speed instead gives less time. Package prices may also fail to scale with volume. Cross multiplication cannot make an unsuitable direct-proportion model valid.

Retain the context checks from the activities: recipe scaling uses the full intended number of servings; price comparisons use equal amounts; brick estimates use the full built wall length with matching height and thickness; approximate annual distances give approximate weekly distances.

7.5 Sharing, but Not Equally!

From Lesson 5: when a whole x is split in ratio m : n, add the ratio terms. There are m+n equal parts, so one part is x/(m+n). Multiply by m or n to find each share.

For 42 counters in ratio 4 : 3, there are seven parts of six counters each. The shares are 24 and 18. Verify both their sum and their ratio. The denominator for a fraction of the whole is m+n, not just the other ratio term.

For changing mixtures, first calculate the actual component amounts. Then identify which stays fixed. In 40 kg of 3 : 1 sand and cement, the amounts are 30 kg and 10 kg. To reach 5 : 2 by adding cement, keep sand at 30 kg, find required cement 12 kg, and add only 2 kg.

An added whole container is not necessarily one ratio part. In a full bucket of 3 : 5 red and yellow paint, the bucket contains eight ratio parts. Adding another equally sized full bucket of yellow gives 3 : 13. Confirm that container capacities match.

A measurement may have fractional parts, but indivisible objects may prevent an exact whole-object split. Interpret the arithmetic according to what is being shared.

7.6 Unit Conversions

From Lesson 6: a conversion changes the unit description of a fixed amount. Predict whether the numerical count should increase or decrease, then use a factor for the correct kind of measurement. Area conversion must use square units, not the length factor alone.

The factors written with ≈ are rounded. Use an approximate answer when they are used. A litre is a volume unit and is not automatically a kilogram of every substance.

Temperature conversion has a fixed offset: 0°C corresponds to 32°F. Therefore Celsius and Fahrenheit readings are not directly proportional. In the reverse formula, subtract 32 before multiplying by 5/9.

Figure it Out

From Lesson 7: first choose the kind of reasoning. Juice comparisons use ratio simplification; buses use capacity per bus and rounding up; crowding uses people per area; the crane example uses a fraction of the whole; and the saffron example uses a fractional scale factor.

The age question distinguishes final age from elapsed years. The given gold-to-water exercise compares masses at equal volume. Farming and land-cost questions require area conversions; the tap question requires volume conversion and constant flow. The tractor question compares time for a fixed job.

The coin challenge combines three operations: divide total mass in a ratio, convert grams to kilograms, and multiply each mass by its corresponding price before adding. Use the stipulated data as a model and do not confuse material cost with monetary value.

Binairo

After the revision, try this logic activity, also called Takuzu. Fill each empty cell with a horizontal line H or a vertical line V. In these six-by-six grids, every row and every column must contain exactly three of each symbol.

Do not put three identical symbols consecutively in a row or column. Every completed row must differ from every other row, and every completed column must differ from every other column. A row is allowed to match a column: the uniqueness comparisons are within rows and within columns.