**Middle School Number Sense Lesson 77: Substitution (Ratio)**

Today's concept was pretty common a few years ago, but it has shown up less often in recent years. It requires a bit of mental gymnastics, but once you learn the pattern, it shouldn't take more than a few seconds to solve. It appeared only

**twice**last year, with a median placement at

**question # 74**.

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It will probably be easier to explain this concept by first showing an example:

**If 4 a's = 5 b's and 2 b's = 8 c's, then 1 a = ___ c's**

**1 a**, and we have several ratios given, let's give it a shot:

- If 4 a's = 5 b's, that means 1 a =
**5/4**b's. - If 2 b's = 8 c's, that means 5/4 b's = (8/2) x 5/4 = (8 x 5)/(2 x 4) = 40/8 =
**5**c's. - Since we know 1 a = 5/4 b's, that means 1 a =
**5**c's.

- Divide by 4: 1/4
- Multiply by 5: 5/4
- Divide by 2: 5/8
- Multiply by 8:
**5**

- Multiply 8 by 5 to get
**40**. - Divide by 2 and 4 to get
**5**.

So to make the calculations standard, let's think of these problems as:

**If W a's = X b's and Y b's = Z c's, then 1 a = ___ c's**

**ZX**

----

YW

----

YW

**Example 1: If 5 a's = 8 b's and 4 b's = 3 c's, then 1 a = ___ c's**

- The formula is ZX/YW, or (3 x 8)/(4 x 5).
- The answer is
**24/20**, which reduces to**6/5**.

**Example 2: If 12 a's = 7 b's and 2 b's = 3 c's, then 1 a = ___ c's**

- The formula is ZX/YW, or (3 x 7)/(2 x 12).
- The answer is
**21/24**, which reduces to**7/8**.

**Example 3: If 4 a's = 8 b's and 3 b's = 2 c's, then 1 a = ___ c's**

- The formula is ZX/YW, or (2 x 8)/(3 x 4).
- The answer is
**16/12**, which reduces to**4/3**.

**Example 4: If 2 a's = 9 b's and 4 b's = 7 c's, then 1 a = ___ c's**

- The formula is ZX/YW, or (7 x 9)/(4 x 2).
- The answer is
**63/8**.

**Example 5: If 2 a’s = 11 b’s and 3 b’s = 18 c’s, then a = ___ c’s**

- The formula is ZX/YW, or (18 x 11)/(3 x 2).
- The answer is 18/6 x 11, or 3 x 11 =
**33**.

**Check back soon for a free worksheet to help you practice SubstRatio.**

**Up Next for Middle School: ExponentFrac**