If x : y = 3 : 2 and y : z = 5 : 4, find the value of x in the ratio x : y : z

Analyze the question: We are given two ratios that share the term y. To combine them into one ratio x : y : z, we make the value of y the same in both.

Concept: To join a : b and b : c, find the LCM of the two values of b. Then scale each ratio so b matches.

Solution:

• x : y = 3 : 2
• y : z = 5 : 4

The y values are 2 and 5. Their LCM is 10.

• Multiply x : y by 5: x : y = 15 : 10
• Multiply y : z by 2: y : z = 10 : 8

Now y = 10 in both, so:

x : y : z = 15 : 10 : 8

The value of x in this ratio is 15.

Check:

• x : y = 15 : 10 = 3 : 2 ✓
• y : z = 10 : 8 = 5 : 4 ✓

Tip: When two ratios share a common term, cross-multiply to equalize it. Here, x gets multiplied by 5 (the y-value from the other ratio), and z by 2 (the y-value from the first ratio). Always check by simplifying your final ratio back to the originals.