4 times the sum of a number y and 7 is equal to 44. Which equation represents this situation?

1. TRANSLATE the problem information piece by piece

• Given statement: "4 times the sum of a number y and 7 is equal to 44"

• Let's break this down systematically:
  • "a number y" → simply \(\mathrm{y}\)
  • "the sum of a number y and 7" → \(\mathrm{y + 7}\)
  • "4 times the sum..." → \(\mathrm{4(y + 7)}\)
  • "...is equal to 44" → \(\mathrm{= 44}\)

2. INFER why parentheses are crucial

• The phrase "4 times the sum" tells us that 4 multiplies the entire result of adding \(\mathrm{y}\) and 7
• Without parentheses, \(\mathrm{4y + 7}\) would mean "4 times y, then add 7"
• With parentheses, \(\mathrm{4(y + 7)}\) means "first find the sum of \(\mathrm{y}\) and 7, then multiply that whole result by 4"

3. Combine all parts to form the equation

• Putting it all together: \(\mathrm{4(y + 7) = 44}\)

Answer: B. \(\mathrm{4(y + 7) = 44}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often translate "4 times the sum of y and 7" as \(\mathrm{4y + 7}\) instead of \(\mathrm{4(y + 7)}\).

They read left to right and translate as they go: "4 times... y" becomes \(\mathrm{4y}\), then "and 7" becomes \(\mathrm{+7}\), giving them \(\mathrm{4y + 7}\). They miss that "4 times" applies to the entire sum, not just to \(\mathrm{y}\).

This leads them to select Choice A (\(\mathrm{4y + 7 = 44}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Some students get confused about the order of operations and write the numbers in the same sequence they hear them.

They might think "4 times... y and 7" and write \(\mathrm{4 + y + 7}\), completely missing that "times" indicates multiplication and "sum" indicates the \(\mathrm{y}\) and 7 should be added first.

This may lead them to select Choice D (\(\mathrm{4 + y + 7 = 44}\)).

The Bottom Line:

The key challenge is recognizing that "4 times the sum" means the multiplication applies to the entire sum operation, requiring parentheses to group the sum before multiplying.