Number System Fundamentals • Topic 7 of 7

Number Word Problems

Word problems are about translation: name the unknown, turn each sentence into an equation, then solve. For sum-and-difference problems, the larger number is (sum + difference)/2 and the smaller is (sum − difference)/2. For digit problems, a two-digit number is 10t + u, its reverse is 10u + t, and their difference is 9(t − u). Consecutive integers are n, n+1, n+2. Watch the wording: "more than" means add, "twice" means multiply by 2, "exceeds" means is greater than. The SAT rewards fast, accurate translation far more than heavy algebra.

✅ Solved examples

1. The sum of two numbers is 25 and their difference is 13. What is the larger number?
The larger number is (sum + difference)/2 = (25 + 13)/2 = 38/2 = 19.
2. The sum of three consecutive integers is 72. What is the largest?
The middle integer is 72 ÷ 3 = 24, so the integers are 23, 24, 25. The largest is 25.
3. A two-digit number has digit sum 9. Adding 27 reverses its digits. Find the number.
Let the number be 10t + u with t + u = 9. Then (10t + u) + 27 = 10u + t gives 9(u − t) = 27, so u − t = 3. Solving with t + u = 9 gives t = 3, u = 6 — the number is 36.
4. A number is 4 more than twice another, and their sum is 31. Find the numbers.
Let the smaller be x; the other is 2x + 4. Then x + (2x + 4) = 31, so 3x = 27 and x = 9. The numbers are 9 and 22.

✏️ Practice — try these, take hints as needed

1. The sum of two numbers is 40 and their difference is 8. What is the smaller number?
Smaller number = (sum − difference)/2.
Substitute 40 and 8.
(40 − 8)/2.
16.
2. The sum of three consecutive integers is 96. What is the middle one?
The three integers are n−1, n, n+1.
Their sum is 3n.
Divide 96 by 3.
32.
3. A two-digit number has digits summing to 11. Reversing it increases the value by 27. Find the number.
Let it be 10t + u with t + u = 11.
Reversed minus original = 9(u − t) = 27, so u − t = 3.
Solve t + u = 11 and u − t = 3.
47.
4. The larger of two numbers is twice the smaller and their sum is 27. Find the larger number.
Let the smaller be x; the larger is 2x.
Then x + 2x = 27.
Solve for x, then double it.
18.
5. A number exceeds one-third of itself by 30. Find the number.
Let the number be x; one-third of it is x/3.
x − x/3 = 30 means (2/3)x = 30.
Multiply both sides by 3/2.
45.

📝 Topic test — 8 questions

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