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Ratio & Proportion Questions, Full Concept How to find Ratios with Tips & Tricks in seconds #hechat
Wiz Odyssey
Overview
This video explains the concepts of ratios and proportions, demonstrating various methods to solve problems involving them. It covers how to combine multiple ratios to find a compound ratio (e.g., A:B:C or A:B:C:D) using two primary techniques: filling in gaps and making common terms equal. The video also delves into the definition and application of proportions, including finding the fourth proportional and solving for unknown values. Several example problems are worked through, illustrating how to apply these methods to different scenarios, including those with decimals and algebraic expressions.
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Chapters
- To find a compound ratio like A:B:C when given A:B and B:C, ensure the common term (B) has the same numerical value in both ratios.
- Method 1 (Gap Filling): Write ratios vertically, fill empty spaces by repeating the adjacent number, then multiply vertically.
- Method 2 (Equalizing Common Term): Find the least common multiple (LCM) of the common term's values in each ratio and adjust the ratios by multiplication so the common term is equal, then combine.
- The resulting ratio cannot be simplified if the numbers do not share common factors.
Understanding how to combine ratios is crucial for problems where relationships between multiple quantities are described sequentially, allowing for a unified representation.
Given A:B = 2:3 and B:C = 4:5, make B equal by multiplying the first ratio by 4 and the second by 3, resulting in A:B = 8:12 and B:C = 12:15. This combines to A:B:C = 8:12:15.
- To find a compound ratio like A:B:C:D from A:B, B:C, and C:D, extend the gap-filling method.
- Write all given ratios vertically, aligning common terms.
- Fill empty spaces by repeating the nearest number to the left or right.
- Multiply the numbers in each column vertically to get the final compound ratio.
This method systematically extends the logic for combining three ratios to any number of sequential ratios, enabling complex relationships to be represented.
Given A:B=2:3, B:C=4:5, C:D=6:7. Write them vertically. Fill gaps: 3,3,5, 4,4,6. Multiply columns: (2*4*6):(3*4*6):(3*5*6):(3*5*7) = 48:72:90:105.
- A proportion states that two ratios are equal (e.g., A:B = C:D).
- In a proportion A:B :: C:D, the 'means' are B and C, and the 'extremes' are A and D.
- The fundamental rule of proportions is that the product of the means equals the product of the extremes (B*C = A*D).
- This rule can be used to find an unknown value (often represented by X) in a proportion.
Proportions are essential for solving problems where quantities are directly related, allowing us to find unknown values based on known relationships.
If A:B = C:X and we know A, B, and C, we can find X using the rule: B*C = A*X. Rearranging gives X = (B*C)/A.
- Problems may involve finding the fourth proportional, where three numbers are given and the fourth is unknown (e.g., a:b = c:x).
- When combining ratios, if a number appears in every ratio, it can be cancelled out to simplify calculations.
- When a ratio is given (e.g., A:B = 2:3), it represents a relationship, not necessarily the exact values; actual values might be 2k and 3k for some factor k.
- Problems can involve algebraic expressions within ratios, requiring substitution and simplification.
These examples demonstrate practical application of proportion rules, including handling common factors and algebraic terms, which are common in real-world problems.
To find the fourth proportional to 3, 7, and 9, set up the proportion 3:7 = 9:x. Cross-multiply to get 3x = 7*9, so x = 63/3 = 21.
- When solving proportions with decimals, convert decimals to fractions to simplify calculations.
- The rule of means and extremes (product of means equals product of extremes) applies universally, even with decimals or fractions.
- Problems might ask for the value to be added to ratio terms to achieve a new ratio; this is solved by setting up an equation with an unknown variable representing the added value.
This section addresses how to handle non-integer values and specific types of ratio problems, expanding the applicability of ratio and proportion concepts.
To find what number to add to 7:11 to get 3:4, set up the equation (7+x)/(11+x) = 3/4. Cross-multiplying gives 4(7+x) = 3(11+x), leading to x=5.
- Problems can involve finding the ratio of combined algebraic terms (e.g., (8x+9y)/(8x+2y)) given the ratio of x:y.
- Substitute the ratio values (e.g., x=5, y=2 if x:y=5:2) into the expression and simplify.
- When given ratios like a:b = 1/2 : 1/3 : 1/4, find the LCM of the denominators (2, 3, 4, which is 12) and multiply each fraction by it to get the simplest integer ratio (6:4:3).
- Some problems require careful interpretation of given information, like distinguishing between sequential ratios (A:B, B:C) and non-sequential ones (A:B, A:D).
These examples showcase how ratio and proportion principles are applied in more complex algebraic and fractional contexts, requiring careful manipulation and understanding of underlying concepts.
If x:y = 5:2, to find the ratio (8x+9y):(8x+2y), substitute x=5k, y=2k: (8(5k)+9(2k))/(8(5k)+2(2k)) = (40k+18k)/(40k+4k) = 58k/44k = 58/44 = 29/22.
Key takeaways
- Ratios represent a comparison between quantities, while proportions state the equality of two ratios.
- Combining sequential ratios requires making the common term numerically identical across ratios.
- The 'gap-filling' method and the 'equalizing common term' method are two effective strategies for combining ratios.
- The product of the means equals the product of the extremes is the core principle for solving proportion problems.
- When dealing with ratios, remember that the numbers often represent a simplified relationship, not exact quantities, unless specified.
- Algebraic expressions within ratios can be solved by substitution and simplification.
- Converting decimals and fractions to a common format (like integers via LCM) simplifies complex ratio calculations.
Key terms
RatioProportionCompound RatioMeansExtremesFourth ProportionalLCM (Least Common Multiple)SimplificationCross-multiplication
Test your understanding
- How can you combine the ratios A:B = 1:2 and B:C = 3:4 into a single ratio A:B:C?
- Explain the difference between the 'gap-filling' method and the 'equalizing common term' method for combining ratios.
- What is the fundamental rule used to solve problems involving proportions?
- How would you find the fourth proportional to the numbers 4, 6, and 10?
- If a ratio is given as 1/3 : 1/5, how would you convert it into a ratio of whole numbers?