Find A:B:C:D when A:B = 2:3 ; B:C = 7:9 ; C:D = 5:7

Options

A. a. 70 : 105 : 135 : 189
B. b. 105 : 115 : 236 :189
C. c. 70 : 124 : 155 : 201
D. d. 12 : 78 : 256 : 189

Answer

To find the ratio A:B:C:D given the ratios A:B = 2:3, B:C = 7:9, and C:D = 5:7, we can follow these steps:

### Step 1: Write down the ratios
We have the following ratios:
1. A:B = 2:3
2. B:C = 7:9
3. C:D = 5:7

### Step 2: Express each variable in terms of a common variable
Let's express A, B, C, and D using a common variable.

From A:B = 2:3, we can set:
- A = 2x
- B = 3x

From B:C = 7:9, we can express C in terms of B:
- B = 7y
- C = 9y

From C:D = 5:7, we can express D in terms of C:
- C = 5z
- D = 7z

### Step 3: Find a common value for B, C
Since B appears in both the first and second ratios, we can equate the two expressions for B:
3x = 7y
From this, we can express y in terms of x:
y = (3/7)x

Now substitute y into the expression for C:
C = 9y = 9 * (3/7)x = (27/7)x

### Step 4: Find a common value for C, D
Now we can equate the two expressions for C:
(27/7)x = 5z
From this, we can express z in terms of x:
z = (27/35)x

Now substitute z into the expression for D:
D = 7z = 7 * (27/35)x = (189/35)x

### Step 5: Write all variables in terms of x
Now we have:
- A = 2x
- B = 3x
- C = (27/7)x
- D = (189/35)x

### Step 6: Find a common denominator
To express A, B, C, and D in a simple ratio, we can find a common denominator. The least common multiple of the denominators (1, 1, 7, 35) is 35.

Now, we can express each variable with a denominator of 35:
- A = 2x = (70/35)x
- B = 3x = (105/35)x
- C = (27/7)x = (135/35)x
- D = (189/35)x

### Step 7: Write the final ratio
Now we can write the ratio A:B:C:D:
A:B:C:D = 70:105:135:189

### Final Step: Simplify the ratio if necessary
The ratio 70:105:135:189 is already in its simplest form.

### Final Answer:
A:B:C:D = 70:105:135:189

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