Ratio, fractions and proportional comparison

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Apuntes de la lección Diapositivas Preguntas de práctica

Apuntes de la lección

Ratios and Fractions of a Whole

A part-to-part ratio A:B = 2:3 compares A directly with B: A/B = 2/3.
The whole is the sum of the parts: for 2:3, the whole is 2+3 = 5 parts.
To write a ratio as a fraction of the whole, put each part over the total: 2:3 gives 2/5 and 3/5.
Do not confuse A/B with A/(A + B): these use different wholes. In A:B = 2:3, A is 2/3 of B but 2/5 of their combined whole.
Write fractions of the same whole with a common denominator, then compare their numerators: 2/5 and 3/5 give ratio 2:3.
When a ratio is given as a:b, the fraction of the whole for the first quantity is a/(a+b).

Compare A with B or with the whole

Expressing One Quantity as a Fraction or Multiple of Another

To express A as a fraction of B, write A/B and simplify if possible.
To express A as a multiple of B, compute A ÷ B; if A = 3B, then A is 3 times B.
Check units: both quantities must be in the same units before comparing.
Example: 20 cm as a fraction of 1 m: convert 1 m to 100 cm, so 20/100 = 1/5.
If A is 3 times B, then B is 1/3 of A; do not invert the comparison.
The result can be a percentage: multiply the fraction by 100%.

Proportion as Equality of Ratios

Two ratios are in proportion if they are equal, e.g. 2:3 = 4:6.
To solve for an unknown in a proportion, set the ratios equal and cross-multiply.
Example: if x:5 = 3:2, then 2x = 15, so x = 7.5.
Proportion is multiplicative. Adding the same amount to both entries does not generally preserve their ratio.
A proportion can be written as an equation of fractions: a/b = c/d.
Equivalent ratios can be found by multiplying or dividing both parts by the same non-zero number.

Relating Ratios to Linear Functions

A ratio a:b can be expressed as a linear function: if y = kx, then y:x = k:1.
The constant of proportionality k is the ratio y/x.
Graphically, proportional relationships are straight lines through the origin.
If two quantities are in proportion, doubling one doubles the other.
This links to direct proportion: y = kx, where k is the constant of proportionality.
Inverse proportion (y = k/x) is not the same as a ratio comparison.

Constant ratio y:x = 3:2

Comparing Lengths, Areas and Volumes Using Scale Factors

For similar shapes, the linear scale factor (LSF) is the ratio of corresponding lengths.
The area scale factor (ASF) is the square of the linear scale factor: ASF = LSF².
The volume scale factor (VSF) is the cube of the linear scale factor: VSF = LSF³.
Example: if LSF = 2, areas are 4 times larger and volumes are 8 times larger.
Given ASF, find LSF by taking the square root; given VSF, take the cube root.
Do not apply the linear factor directly to areas or volumes.

From a linear factor to area and volume

Solving Problems with Similar Figures

Identify corresponding lengths, areas, or volumes in similar figures.
Write the ratio of corresponding lengths to find the linear scale factor.
Use the scale factor to find missing lengths, areas, or volumes.
If the area scale factor is 9, the linear scale factor is 3 and the volume scale factor is 27.
If the volume scale factor is 64, the linear scale factor is 4 and the area scale factor is 16.
Always state units and check that the answer makes sense (e.g. larger shape has larger area).

Common Misconceptions and Errors

Confusing 2/3 of the second quantity with 2/5 of the combined whole in a part-to-part ratio 2:3.
Applying the linear scale factor to areas or volumes instead of squaring or cubing.
Inverting a multiplicative comparison: if A is 3 times B, then B = A/3, not 3A.
Treating proportion as additive: adding the same amount to both quantities does not preserve the ratio.
Comparing quantities in different units without converting first (e.g. cm vs m).
Forgetting to find the whole when converting a ratio to fractions.

Typical Exam Question Forms

Express a ratio as a fraction of the whole and vice versa.
Given the area scale factor of two similar shapes, find the linear or volume scale factor.
Express one quantity as a percentage or fraction of another.
Show that two ratios are equivalent by simplifying or cross-multiplying.
Solve a problem comparing the volumes of two similar solids using scale factors.
Use proportion to find an unknown quantity in a real-life context.

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Preguntas de práctica

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1.The ratio of oranges to apples in a fruit bowl is 2:3. What fraction of the fruit are oranges?
Easy
- A2/3
- B2/5
- C3/5
- D3/2
2.A model car is built to a scale of 1:20. The length of the model is 12 cm. What is the length of the real car in metres?
Medium
- A2.4 m
- B24 m
- C0.6 m
- D240 m
3.Two similar rectangles have a linear scale factor of 3. The area of the smaller rectangle is 5 cm². What is the area of the larger rectangle?
Medium
- A15 cm²
- B45 cm²
- C75 cm²
- D125 cm²
4.Which of the following statements are true? (Select all that apply)
Medium
- AIf A is three times B, then B is one third of A.
- BIf the ratio of x to y is 2:3, then x is 2/3 of y.
- CIf two shapes are similar with linear scale factor k, their areas scale by k².
- DIf two quantities are in proportion, increasing one by a fixed amount increases the other by the same amount.
5.For two shares compared by a part-to-part ratio: The ratio 2:3 means the same as the fraction 2/3 of the whole.
Easy
True or false?
6.Match each ratio to its equivalent fraction of the whole.
Medium
- 2:3
- 3:2
- 1:4
- 2/5
- 3/5
- 1/5
7.Order these quantities from smallest to largest numerical value.
Medium
- 3/5
- 0.65
- 70%
- 2/5
8.Two cuboids are similar with linear scale factor 2 from smaller to larger. The smaller has surface area 40 cm². What is the larger surface area?
Medium
- A80 cm²
- B160 cm²
- C320 cm²
- D640 cm²

Más preguntas

9.Match each ratio to the fraction of the whole it represents.
Easy
- 1:3
- 2:5
- 3:4
- 1/4
- 2/7
- 3/7
10.Order these ratios from smallest to largest fraction of the whole represented by the first part.
Medium
- 1:5
- 2:3
- 3:4
- 1:2
11.A and B are two similar squares. The side length of A is 3 cm and the side length of B is 6 cm. Which of the following statements are true? (Select all that apply)
Medium
- AThe linear scale factor from A to B is 2.
- BThe area scale factor from A to B is 2.
- CThe area of B is 4 times the area of A.
- DThe perimeter of B is 2 times the perimeter of A.
- EThe area of A is half the area of B.
12.The ratio of boys to girls in a school is 5:7. What fraction of the students are boys?
Medium
- A5/7
- B5/12
- C7/12
- D7/5
13.Two similar prisms have base areas in the ratio 9:16. The volume of the smaller prism is 54 cm³. What is the volume of the larger prism?
Hard
- A72 cm³
- B96 cm³
- C128 cm³
- D144 cm³
14.The ratio of the lengths of two similar rectangles is 2:3. The area of the smaller rectangle is 20 cm². What is the area of the larger rectangle?
Medium
- A30 cm²
- B45 cm²
- C60 cm²
- D90 cm²
15.Match each pair of similar shapes to the correct scale factor relationship.
Medium
- Lengths 2:5
- Areas 4:9
- Volumes 27:64
- Area scale factor 4:25
- Linear scale factor 2:3
- Linear scale factor 3:4
16.The ratio of cats to dogs in a pet shop is 3:4. What fraction of the animals are cats?
Easy
- A3/4
- B3/7
- C4/7
- D1/3
17.If the ratio of x to y is 2:5, then x is 2/5 of y.
Easy
True or false?
18.The ratio of the areas of two similar circles is 4:9. What is the ratio of their radii?
Medium
- A2:3
- B4:9
- C16:81
- D2:9
19.A map has a scale of 1:50000. Two towns are 8 cm apart on the map. What is the actual distance in kilometres?
Medium
- A4 km
- B40 km
- C0.4 km
- D400 km
20.Match each ratio to the fraction of the whole represented by the first part.
Medium
- 2:3
- 3:5
- 4:7
- 2/5
- 3/8
- 4/11
21.The ratio of the volumes of two similar solids is 27:64. What is the ratio of their corresponding linear dimensions?
Medium
- A3:4
- B9:16
- C27:64
- D√27:8
22.Two similar cylinders have heights in the ratio 2:5. Which of the following statements are true? (Select all that apply)
Hard
- AThe ratio of their base radii is 2:5.
- BThe ratio of their curved surface areas is 4:25.
- CThe ratio of their volumes is 4:25.
- DThe ratio of their total surface areas is 4:25.
- EThe ratio of their volumes is 8:125.
23.If the ratio of x to y is 3:7, then x is 3/7 of the total of x and y.
Easy
True or false?
24.The ratio of the lengths of two similar rectangles is 3:4. The area of the smaller rectangle is 27 cm². What is the area of the larger rectangle?
Medium
- A36 cm²
- B48 cm²
- C64 cm²
- D81 cm²
25.A model car is built to a scale of 1:50. The real car has a length of 4.5 m. Which of the following statements are true? (Select all that apply)
Hard
- AThe model length is 9 cm.
- BThe model length is 0.09 m.
- CThe model length is 90 mm.
- DThe model length is 4.5 cm.
- EThe model length is 450 mm.
26.Match each ratio to the equivalent fraction of the whole represented by the first part.
Medium
- 2:3
- 3:5
- 4:7
- 1:4
- 2/5
- 3/8
- 4/11
- 1/5
27.What does the ratio 3:4 mean in terms of the whole when comparing two parts?
Easy
- A3 parts to 4 parts, making 7 parts in total
- B3 parts to 4 parts, making 4 parts in total
- C3 parts out of 4 parts
- D4 parts out of 3 parts
28.Which of the following is the correct way to express the ratio 2:5 as a fraction of the whole?
Easy
- A2/5
- B2/7
- C5/2
- D5/7
29.The ratio 4:1 means that the first quantity is four times the second quantity.
Easy
True or false?
30.Express 30 cm as a fraction of 1.2 m. Give your answer in its simplest form.
Easy
- A1/4
- B1/3
- C2/5
- D3/4
31.A model car is built to a scale of 1:50. The real car has a length of 4.5 m. What is the length of the model in centimetres?
Medium
- A9 cm
- B0.09 cm
- C90 cm
- D225 cm
32.What is the correct way to compare quantities in a ratio?
Easy
- AConvert to the same units
- BConvert to different units
- CAdd the quantities
- DSubtract the quantities
33.For two shares compared by a part-to-part ratio: The ratio 2:3 is the same as the fraction 2/3 of the whole.
Easy
True or false?
34.If A is twice as big as B, then B is half of A.
Easy
True or false?
35.A and B are two similar triangles. The area of A is 4 cm² and the area of B is 16 cm². Which of the following statements are true? (Select all that apply)
Medium
- AThe linear scale factor from A to B is 2.
- BThe linear scale factor from A to B is 4.
- CThe volume scale factor from A to B is 8.
- DThe ratio of corresponding sides is 1:2.
- EThe ratio of areas is 1:4.
36.For two shares compared by a part-to-part ratio: The ratio 3:5 means that the first quantity is 3/5 of the whole.
Easy
True or false?
37.A map has a scale of 1:100000. Two towns are 3.5 cm apart on the map. What is the actual distance in kilometres?
Medium
- A3.5 km
- B35 km
- C350 km
- D0.35 km
38.The ratio of the lengths of two similar rectangles is 3:5. The area of the smaller rectangle is 27 cm². What is the area of the larger rectangle?
Medium
- A45 cm²
- B75 cm²
- C125 cm²
- D15 cm²
39.A recipe uses flour and sugar in the ratio 5:2. If 350 g of flour is used, how many grams of sugar are needed?
Medium
40.Complete the statement.
Easy
To compare lengths, areas and volumes of similar shapes, use the ____ scale factor for areas and the ____ scale factor for volumes.
41.Two shares have part-to-part ratio 1:3. Write the first share as a fraction of their combined whole.
Easy
42.Two cuboids are similar with linear scale factor 2 from smaller to larger. The smaller volume is 24 cm³. What is the larger volume?
Medium
43.Express 30 cm as a fraction of 1.2 m.
Medium
44.If A = 2B, write the complete ratio A:B in its simplest whole-number form.
Medium
45.A class has 12 boys and 18 girls. What proportion of the students are boys? Give your answer as a decimal.
Easy
46.Express 40 cm as a fraction of 2 m. Give your answer in its simplest form.
Easy
47.The ratio of the lengths of two similar rectangles is 2:5. The area of the smaller rectangle is 12 cm². Find the area of the larger rectangle.
Medium
48.A map has a scale of 1:25000. Two towns are 6 cm apart on the map. Find the actual distance in kilometres.
Medium
49.A map has a scale of 1:20000. Two towns are 7.5 cm apart on the map. What is the actual distance in kilometres?
Medium
50.Express 250 g as a fraction of 1.5 kg. Give your answer in its simplest form.
Medium
51.Explain why the ratio 2:3 is not equivalent to the fraction 2/3 of a whole.
Medium
52.Two positive quantities have A:B = 2:3. Explain why A is 2/3 of B but 2/5 of their combined whole.
Medium
53.The volumes of two similar solids are in the ratio 8:27. Find the ratio of their surface areas.
Hard
54.A rectangle is 12 cm long and 8 cm wide. A similar larger rectangle has area 216 cm². Find its length.
Hard
55.A recipe uses flour and sugar in the ratio 5:2. If 350 g of flour is used, how many grams of sugar are needed?
Medium
- A154 g
- B168 g
- C126 g
- D140 g
56.Two cuboids are similar with linear scale factor 2 from smaller to larger. The smaller volume is 24 cm³. What is the larger volume?
Medium
- A211 cm³
- B192 cm³
- C230 cm³
- D173 cm³
57.A class has 12 boys and 18 girls. Express the number of boys as a fraction of the total number of students. Give your answer in its simplest form.
Easy
- A\frac{1}{5}
- B\frac{3}{5}
- C\frac{2}{5}
- D\frac{2}{3}
58.A map has a scale of 1:20000. Two towns are 7.5 cm apart on the map. What is the actual distance in kilometres?
Medium
- A3.5 km
- B0.5 km
- C2.5 km
- D1.5 km

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