Class 6 Maths Ratio and Proportion Notes

Introduction:
The chapter “Ratio and Proportion” introduces students to the concepts of ratio and proportion, which are essential in comparing quantities and understanding their relationships. Whether you’re dividing a pizza, sharing money, or analyzing recipes, ratios and proportions help in comparing amounts and finding equivalent values. This chapter will help students understand how to express relationships between numbers and how to solve problems involving proportionality.

Key Concepts Covered:

1. What is a Ratio?
   • A ratio is a way of comparing two quantities of the same kind. It shows how many times one number contains another.
   • A ratio is written in the form $\frac{a}{b}$ba​ or a : b, where $a$a and $b$b are the two quantities being compared.
   • Example: If there are 4 boys and 5 girls in a class, the ratio of boys to girls is $\frac{4}{5}$54​ or 4:5.
2. Types of Ratios:
   • Ratio of Like Quantities: Ratios that compare quantities of the same type (e.g., comparing number of boys to number of girls).
   • Ratio of Unlike Quantities: Ratios that compare different types of quantities (e.g., comparing the cost of 5 pencils to the total weight of 2 books).
3. Simplifying Ratios:
   • Just like fractions, ratios can be simplified by dividing both terms by their Greatest Common Divisor (GCD).
   • Example: The ratio 6:8 can be simplified to 3:4 by dividing both 6 and 8 by 2 (their GCD).
4. What is Proportion?
   • Proportion is an equation that shows two ratios are equivalent. If two ratios are equal, we say they are in proportion.
   • Example: If $\frac{a}{b} = \frac{c}{d}$ba​=dc​, then $a : b :: c : d$a:b::c:d (read as “a is to b as c is to d”).
5. Types of Proportions:
   • Direct Proportion: Two quantities are said to be in direct proportion if when one quantity increases, the other increases by the same factor, or when one decreases, the other decreases.
     • Example: If the number of workers increases, the total work done also increases proportionally.
   • Inverse Proportion: Two quantities are in inverse proportion if one quantity increases while the other decreases.
     • Example: The more people there are to complete a job, the less time it takes.
6. Solving Proportions:
   • To solve a proportion, use the cross-multiplication method.
   • Example: If $\frac{3}{4} = \frac{x}{8}$43​=8x​, cross-multiply to get:
     $3 \times 8 = 4 \times x$3×8=4×x
     $24 = 4x$24=4x
     $x = 6$x=6.

Important Questions with Answers:

1. What is the ratio of 12 apples to 16 oranges?
   • Answer:
     $\frac{12}{16} = \frac{3}{4}$1612​=43​
     So, the simplified ratio is 3:4.
2. If the ratio of boys to girls in a class is 5:6, and there are 30 boys, how many girls are there?
   • Answer:
     $\frac{5}{6} = \frac{30}{x}$65​=x30​, where $x$x is the number of girls.
     Cross-multiply:
     $5x = 180$5x=180
     $x = \frac{180}{5} = 36$x=5180​=36.
     So, there are 36 girls.
3. If 23=x6\frac{2}{3} = \frac{x}{6}32​=6x​, find the value of xxx.
   • Answer:
     Cross-multiply:
     $2 \times 6 = 3 \times x$2×6=3×x
     $12 = 3x$12=3x
     $x = \frac{12}{3} = 4$x=312​=4.
4. The ratio of boys to girls in a class is 7:8. If there are 49 boys, how many girls are there?
   • Answer:
     $\frac{7}{8} = \frac{49}{x}$87​=x49​, where $x$x is the number of girls.
     Cross-multiply:
     $7x = 392$7x=392
     $x = \frac{392}{7} = 56$x=7392​=56.
     So, there are 56 girls.
5. If the number of workers and the time taken to finish a task are in inverse proportion, and 5 workers take 10 days to complete a job, how many days will 10 workers take?
   • Answer:
     Since workers and time are in inverse proportion,
     $5 \times 10 = 10 \times x$5×10=10×x
     $50 = 10x$50=10x
     $x = \frac{50}{10} = 5$x=1050​=5.
     So, 10 workers will take 5 days.
6. Simplify the ratio 15:25.
   • Answer:
     The GCD of 15 and 25 is 5, so divide both terms by 5:
     $\frac{15}{25} = \frac{3}{5}$2515​=53​.
     The simplified ratio is 3:5.