Understanding Ratios Ratios compare two quantities of the same kind. For example, comparing 5 kg books to 20 kg men is valid because both are in kilograms, but comparing them to liters of milk isn't meaningful. Simplifying ratios involves finding common factors; for instance, simplifying a ratio like 12:16 by dividing through by their greatest common divisor.
Defining and Comparing Ratios To determine which ratio is greater or smaller without calculators, make denominators equal using multiplication. This allows direct comparison based on numerators alone. Multiplying each term in a fraction equally doesn't change its value but helps convert fractions into integers for easier calculations.
Simplification Using LCMs When working with fractional ratios such as 1/3:1/8:1/6, find the least common multiple (LCM) of all denominators to simplify them into whole numbers while maintaining proportionality between terms.
'Dividing Quantities Based on Ratio' Examples 'If ten books are divided between Ram and Sham at a given proportion (e.g., '2 parts vs three'), calculate individual shares accordingly.' Similarly applies when scaling up/down total items distributed across predefined proportions among participants involved therein!
Coffee and Cocoa Mixture Ratio Problem A coffee mixture problem involves replacing a portion of coffee with cocoa powder. Initially, 25 grams of coffee is reduced by removing 5 grams and adding an equal amount of cocoa powder. This process repeats, leaving the final ratio between cocoa in mixtures P (initial) to Q (final) as 5:9.
Speed-Time-Distance Basics The fundamental formulas for speed-time-distance are reviewed: Speed = Distance/Time; Time = Distance/Speed; Distance = Speed × Time. Problems introduce concepts like relative speed through examples involving cars covering distances at different speeds or trains traveling between stations under varying conditions.
Relative Speeds Between Two Bodies Two bodies moving at speeds in the ratio 2:3 cover a distance where one takes longer than the other due to its slower pace. Calculations show how their time difference relates directly to their respective velocities using basic algebraic manipulation.
'Usual' vs 'Altered' Travel Times Analysis When altering travel speeds—either faster or slower—the resulting changes in arrival times can be calculated based on constant distances covered during journeys from city X to Y under various scenarios such as increased/decreased percentages affecting overall timing outcomes accordingly
'Double Journey’ Scenario With Increased Velocity Impact. . A car initially travels certain kilometers within specific hours but later doubles this journey length while increasing velocity percentage-wise leading towards recalculating new timings required fulfilling extended trip demands successfully without errors involved therein .