If [tex]$X: Y=4: 7$[/tex] and [tex]$Y: Z=5: 3$[/tex], find the ratio [tex]$X: Y: Z$[/tex].
Answer (2)
Write the given ratios as fractions: Y X = 7 4 and Z Y = 3 5 .
Find the least common multiple (LCM) of the Y values (7 and 5), which is 35.
Adjust the ratios to have the same Y value: Y X = 35 20 and Z Y = 21 35 .
Combine the ratios to get the final answer: X : Y : Z = 20 : 35 : 21 .
Explanation
Understanding the Problem We are given two ratios: X : Y = 4 : 7 and Y : Z = 5 : 3 . Our goal is to find the combined ratio X : Y : Z . To do this, we need to make the Y values in both ratios the same.
Writing Ratios as Fractions First, write the given ratios as fractions: Y X = 7 4 and Z Y = 3 5 .
Finding the Least Common Multiple (LCM) To combine the ratios, we need to find a common value for Y . The least common multiple (LCM) of the Y values, which are 7 and 5, is 35.
Adjusting the First Ratio Multiply the first ratio by 5 5 to get an equivalent ratio with Y = 35 :
Y X = 7 × 5 4 × 5 = 35 20 .
Adjusting the Second Ratio Multiply the second ratio by 7 7 to get an equivalent ratio with Y = 35 :
Z Y = 3 × 7 5 × 7 = 21 35 .
Combining the Ratios Now we have Y X = 35 20 and Z Y = 21 35 . Since the Y values are the same, we can combine the ratios to find X : Y : Z .
Final Answer Therefore, the combined ratio is: X : Y : Z = 20 : 35 : 21.
Examples
Ratios are incredibly useful in everyday life. Imagine you're baking a cake and the recipe calls for a ratio of flour to sugar of 2:1. If you want to scale up the recipe, you need to maintain this ratio to ensure the cake turns out right. Similarly, ratios are used in mixing paints, calculating financial ratios, and even in determining the gear ratios in a bicycle to optimize speed and effort. Understanding ratios helps in making informed decisions and achieving desired outcomes in various practical situations.
The combined ratio of X : Y : Z is 20 : 35 : 21 . This ratio is obtained by adjusting the initial ratios to have a common Y value of 35. The process involves multiplying each ratio appropriately to achieve this commonality before combining them.
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