Problem 1: Convert to symbols: "The sum of a number $x$ and 5 is equal to 10."

Solution:

• "sum of $x$ and 5" = $x + 5$
• "is equal to" = $=$
• "10" = $10$

Answer: $x + 5 = 10$

Problem 2: Convert to symbols: "For every number $x$, if $x$ is greater than 2, then $x$ squared is greater than 4."

Solution:

• "For every" = $\forall$
• "if...then" = $\Rightarrow$
• "$x$ is greater than 2" = $x > 2$
• "$x$ squared is greater than 4" = $x^2 > 4$

Answer: $\forall x > 2, x^2 > 4$ or $\forall x \in \mathbb{R} : (x > 2 \Rightarrow x^2 > 4)$

Problem 3: Convert to symbols: "The limit of function $f$ as $x$ approaches infinity is zero."

Solution:

• "limit" = $\lim$
• "as $x$ approaches infinity" = $x \to \infty$
• "is zero" = $= 0$

Answer: $\lim_{x \to \infty} f(x) = 0$

Problem 4: Convert to symbols: "A is a subset of B, but A is not equal to B."

Solution:

• "A is a subset of B" = $A \subset B$
• "but" = and
• "A is not equal to B" = $A \neq B$

Answer: $A \subset B$ and $A \neq B$ or $A \subsetneq B$ (proper subset)

Problem 5: Convert to symbols: "The integral of $f(x)$ from 0 to 5."

Solution:

• "integral" = $\int$
• "$f(x)$" = $f(x)$
• "from 0 to 5" = limits of integration

Answer: $\int_0^5 f(x) \, dx$

Daily Life Applications

Converting English to mathematical symbols helps you express real-world situations precisely and solve problems systematically.

Budgeting and Finance

• "The total cost is the sum of item prices" → $\text{Total} = \text{price}_1 + \text{price}_2 + \text{price}_3$
• "My savings must be greater than $1000" → $\text{savings} > 1000$
• "Spending should not exceed income" → $\text{spending} \leq \text{income}$

Shopping and Discounts

• "If I spend more than $50, I get 10% off" → $\text{total} > 50 \Rightarrow \text{discount} = 0.10 \times \text{total}$
• "The final price equals the original price minus the discount" → $\text{final} = \text{original} - \text{discount}$

Time Management

• "Total time equals work time plus break time" → $\text{total time} = \text{work} + \text{break}$
• "I need at least 8 hours of sleep" → $\text{sleep} \geq 8$
• "Meeting duration is between 30 and 60 minutes" → $30 \leq \text{duration} \leq 60$

Health and Fitness

• "Target heart rate is between 120 and 150" → $120 \leq \text{heart rate} \leq 150$
• "Calories consumed should equal calories burned" → $\text{calories in} = \text{calories out}$ (for weight maintenance)
• "Weight loss equals calories burned minus calories consumed" → $\text{weight loss} \propto (\text{burned} - \text{consumed})$

Cooking and Recipes

• "If I double the recipe, multiply all ingredients by 2" → If $\text{servings} = 2 \times \text{original}$, then $\text{ingredient} = 2 \times \text{original amount}$
• "The ratio of flour to sugar is 3 to 1" → $\frac{\text{flour}}{\text{sugar}} = \frac{3}{1}$

Travel and Distance

• "Total distance equals speed times time" → $\text{distance} = \text{speed} \times \text{time}$
• "Average speed equals total distance divided by total time" → $\text{average speed} = \frac{\text{total distance}}{\text{total time}}$

Problem-Solving Strategy

When translating English to math:

1. Identify key quantities (what numbers or variables are mentioned?)
2. Find relationships (sum, difference, product, ratio, etc.)
3. Recognize keywords ("sum" = $+$, "difference" = $-$, "times" = $\times$, "per" = $\div$)
4. Use appropriate symbols ($=$, $>$, $<$, $\leq$, $\geq$)
5. Check your translation makes sense in context

Converting everyday language to mathematical symbols helps you solve problems more systematically and communicate ideas more precisely!