Multi-step word problems in 7th grade

Answer. 2.4

The answer is 2.4 because the parentheses equal 0.8 and $3 \times 0.8 = 2.4$.

1. 1Start with the expression inside the parentheses because grouping symbols come first.
2. 2Convert $\frac{1}{2}$ to $0.5$, then add inside the parentheses to get $0.8$.
3. 3Multiply $3$ by $0.8$ to get the final answer.

Common mistake. A common mistake is adding $3 + 0.5 + 0.3$ instead of multiplying. Remember the coefficient 3 multiplies the entire grouped result: the parentheses equal 0.8 first, then $3 \times 0.8 = 2.4$.

The idea underneath. Parentheses make part of an expression act like one number; once you replace that group with its value, the rest of the order of operations is easier to follow.

Match $\frac{3}{5} - 0.1 + 0.05$ with $\frac{1}{2}$, match $-\frac{9}{10} - 0.2 + 0.1$ with $-1$, and match $\frac{3}{10} - 0.4 + 0.1$ with $0$. The estimate $1$ is not used.

1. 1The available estimates are $0$, $\frac{1}{2}$, $-1$, and $1$. For each expression, replace each rational number with a nearby friendly value and decide which benchmark the total is closest to.
2. 2$\frac{3}{5}$ is about $0.6$, so the first expression is approximately $0.6 - 0.1 + 0.05 = 0.55$. The closest estimate is $\frac{1}{2}$.
3. 3Keep the negative sign on $-\frac{9}{10}$. It is about $-0.9$, so the second expression is approximately $-0.9 - 0.2 + 0.1 = -1.0$. The closest estimate is $-1$.
4. 4$\frac{3}{10}$ is about $0.3$, so the third expression is approximately $0.3 - 0.4 + 0.1 = 0$. The closest estimate is $0$.

Common mistake. A common mistake is dropping the negative sign in $-\frac{9}{10}$ and treating the second expression as about $0.8$, then incorrectly matching it to $1$ instead of $-1$.

The idea underneath. A strong estimate comes from replacing each rational number with a nearby friendly value and then deciding which displayed benchmark the total is closest to.

The answer is 1 because the amounts round to -5, 12, and -6, and $-5 + 12 - 6 = 1$, so the balance is about $1.

1. 1Translate the story into signed numbers: the debt is negative, earning is positive, and spending is negative.
2. 2Round each amount to a nearby whole number.
3. 3Combine the rounded numbers.
4. 4So the best estimate is the option showing 1.

Common mistake. A common mistake is dropping the starting debt and estimating only $12.15 - 6.20 \approx 6$ or 7. Always include the opening debt as a negative amount before combining the later changes.

The idea underneath. In signed real-life problems, decide whether each amount adds to or subtracts from the total before you round.

Answer. 15+3\left(4-1.5\right)+\left(-5\right)

One correct calculation is $15 + 3(4 − 1.5) + (−5)$ because it starts at 15, shows 3 grouped rounds of earning 4 and losing 1.5, and then subtracts 5 without solving.

1. 1The starting score is 15.
2. 2Each round has two changes: earning 4 points and losing 1.5 points. Both happen within the same round, so they must be grouped together as (4 − 1.5). Parentheses are required to show that both changes belong to each round.
3. 3That combined change repeats 3 times, so write 3(4 − 1.5).
4. 4After the rounds, the player loses 5 more points. A loss is a negative change, so add −5.
5. 5Put all parts together and stop there, because the problem asks for the calculation only.

Common mistake. A common mistake is writing $15 + 3 \times 4$ − 1.5 + (−5) instead of $15 + 3(4 − 1.5) + (−5)$, which applies the 3 only to the 4-point gain and ignores the 1.5-point loss. Both changes happen every round, so both must be inside the parentheses. Another mistake is simplifying 4 − 1.5 to 2.5 before writing the expression.

The idea underneath. Repeated change is shown by multiplying the number of repeats by the signed change, then combining that with the start and final adjustment.