Exercise practice 1: Proportional Reasoning-2

3.1 Figure it Out: dividing a whole in a given ratio (page 60)

These questions use one method. Add the terms of the ratio, divide the whole by this sum to get one part, and multiply each term by one part. For a triangle, remember that the angles add to 180 degrees and the sides must obey the triangle inequality.

3.4 Question 1: The cricket coach's schedule

Statement. A cricket coach plans warm-up and cool-down : batting : bowling : fielding in the ratio 3 : 4 : 3 : 5. A session lasts 150 minutes. How long is each activity?

Answer. Warm-up and cool-down 30 minutes, batting 40 minutes, bowling 30 minutes, fielding 50 minutes.

Steps.

1. Sum of the terms = 3 + 4 + 3 + 5 = 15 parts.
2. One part = 150/15 = 10 minutes.
3. Warm-up and cool-down = 3 × 10 = 30 minutes.
4. Batting = 4 × 10 = 40 minutes, bowling = 3 × 10 = 30 minutes, fielding = 5 × 10 = 50 minutes.
5. Check: 30 + 40 + 30 + 50 = 150 minutes.

3.4 Question 2: Books in three languages

Statement. A library has Odiya, Hindi and English books in the ratio 3 : 2 : 1. It has 288 Odiya books. Find the number of Hindi and English books.

Answer. Hindi 192 books, English 96 books.

Steps.

1. The Odiya share is 3 parts, and 3 parts = 288 books.
2. One part = 288/3 = 96 books.
3. Hindi = 2 parts = 192 books, English = 1 part = 96 books.
4. Check: 288 + 192 + 96 = 576 books in all, and 288 : 192 : 96 = 3 : 2 : 1.

3.4 Question 3: One hundred coins

Statement. A purse holds 100 coins of four values, ₹10, ₹5, ₹2 and ₹1. Their counts follow the ratio 4 : 3 : 2 : 1. Find the total value.

Answer. ₹600.

Steps.

1. Sum of the terms = 4 + 3 + 2 + 1 = 10 parts.
2. One part = 100/10 = 10 coins.
3. ₹10 coins = 40, ₹5 coins = 30, ₹2 coins = 20, ₹1 coins = 10.
4. Value = 40 × 10 + 30 × 5 + 20 × 2 + 10 × 1 = 400 + 150 + 40 + 10 = ₹600.
5. Check the coin count: 40 + 30 + 20 + 10 = 100 coins.

3.4 Question 4: A triangle with sides 3 : 4 : 5

Statement. Construct a triangle with side lengths in the ratio 3 : 4 : 5. Will every triangle drawn with this side ratio be congruent to the others? Why or why not?

Answer. Yes, such a triangle can be drawn, and it is right angled. No, not all triangles with this ratio are congruent; they are similar, and they become congruent only when the scale factor is the same.

Steps.

1. Pick a scale factor, for example 1 cm per part. The sides are 3 cm, 4 cm and 5 cm.
2. The triangle is constructible because 3 + 4 > 5, so the triangle inequality holds.
3. It is right angled because 3 squared plus 4 squared is 9 + 16 = 25, which equals 5 squared. So the angle between the sides 3 cm and 4 cm is a right angle.
4. Another student may choose 2 cm per part, giving sides 6 cm, 8 cm and 10 cm. Those two triangles have the same shape but different sizes, so they are similar, not congruent.
5. Congruence needs a fixed scale, for example all sides given in centimetres with the same multiplier.

3.4 Question 5: A triangle with sides 1 : 3 : 5

Statement. Can a triangle be drawn with sides in the ratio 1 : 3 : 5? Explain.

Answer. No.

Steps.

1. Take the three sides as k, 3k and 5k for some positive k.
2. The triangle inequality requires the sum of any two sides to be greater than the third.
3. The two shorter sides add to k + 3k = 4k, which is less than the longest side 5k.
4. Since 4k < 5k for every positive k, the shorter sides cannot meet, and no triangle exists with this ratio.

3.2 Figure it Out: a slice of the pie (page 62)

For a pie chart, convert each quantity into an angle by multiplying by 360/(total). Check that the angles add up to 360 degrees before drawing.

3.5 Question 1: Favourite seasons

Statement. 360 people were asked to choose a favourite season. 90 chose summer, 120 chose rainy and the rest chose winter. Draw a pie chart.

Answer. Summer 90 degrees, rainy 120 degrees, winter 150 degrees.

Steps.

1. Winter = 360 - 90 - 120 = 150 people.
2. Summer angle = 90/360 × 360 = 90 degrees.
3. Rainy angle = 120/360 × 360 = 120 degrees.
4. Winter angle = 150/360 × 360 = 150 degrees.
5. Check: 90 + 120 + 150 = 360 degrees. Draw a circle, mark a radius, and measure the angles one after another in the same direction.

3.5 Question 2: Favourite TV channels

Statement. A survey reports favourite channel types: entertainment 50%, sports 25%, news 15% and information 10%. Draw the pie chart.

Answer. 180 degrees, 90 degrees, 54 degrees and 36 degrees respectively.

Steps.

1. The percentages add to 50 + 25 + 15 + 10 = 100%.
2. Convert each percentage to an angle by multiplying by 3.6, since 100% is 360 degrees.
3. Entertainment = 50 × 3.6 = 180 degrees.
4. Sports = 25 × 3.6 = 90 degrees, news = 15 × 3.6 = 54 degrees, information = 10 × 3.6 = 36 degrees.
5. Check: 180 + 90 + 54 + 36 = 360 degrees.

3.5 Question 3: Favourite subjects of your class

Statement. Collect data on the favourite subject of every student in your class, one subject per student, and draw a pie chart.

Answer. The method is fixed, but the numbers depend on your class. The table below shows a worked example for a class of 40 students so the procedure can be checked.

Steps.

1. Write the subject counts in a table and add them to find the total.
2. Simplify the counts into a ratio if possible, then divide 360 degrees by the sum of the terms.
3. Multiply each term by the value of one part.

Sample data (to be replaced by your class data): Mathematics 12 students, Science 8, Social Science 10, Language 6, Arts 4. Total 40 students. 4. One part = 360/40 = 9 degrees per student. 5. Angles: Mathematics 108 degrees, Science 72 degrees, Social Science 90 degrees, Language 54 degrees, Arts 36 degrees. 6. Check: 108 + 72 + 90 + 54 + 36 = 360 degrees. Label the chart with the subject names and counts.