Word problems on pairs of equations

A method for word problems

1. Name the unknowns. Choose a letter for each, and write down what it stands for, with its unit.
2. Write an equation for each fact in the problem.
3. Solve the pair by substitution, elimination or cross-multiplication.
4. Check against the words, not only against the equations. A digit cannot be $12$, and an age cannot be negative.
5. Answer in words, with units.

Numbers and digits

The digits of a two-digit number add up to $11$. If $27$ is added to the number, the digits swap places. Find the number.

Let the tens digit be $x$ and the units digit be $y$. The number is $10x + y$, and with the digits swapped it is $10y + x$.

• The digits add to $11$: $x + y = 11$.
• Adding $27$ swaps them: $10y + x = (10x + y) + 27$, which simplifies to $y - x = 3$.

Add the two equations: $2y = 14$, so $y = 7$ and $x = 4$. The number is $47$. Check. Swapping gives $74$, and $47 + 27 = 74$.

Ages

The sum of the ages of a father and his son is $56$ years. In $8$ years the father will be $3$ times as old as the son. Find their present ages.

Let the father be $f$ years old and the son $s$ years old.

• Their ages add to $56$: $f + s = 56$.
• In $8$ years: $f + 8 = 3(s + 8)$, which simplifies to $f - 3s = 16$.

Subtract the second equation from the first: $4s = 40$, so $s = 10$ and $f = 46$. Check. In $8$ years they will be $54$ and $18$, and $54 = 3 \times 18$.

Speed and distance in a stream

A boat goes $24$ km downstream in $2$ hours and the same distance upstream in $3$ hours. Find the speed of the boat in still water and the speed of the stream.

Let the boat go $u$ km/h in still water and the stream flow at $v$ km/h. Downstream, the stream helps, so the speed is $u + v$. Upstream, it works against the boat, so the speed is $u - v$.

• Downstream: $u + v = \dfrac{24}{2} = 12$.
• Upstream: $u - v = \dfrac{24}{3} = 8$.

Add the equations: $2u = 20$, so $u = 10$. Then $v = 2$. The boat goes $10$ km/h in still water, and the stream flows at $2$ km/h.

Fractions

A fraction becomes $\dfrac{2}{3}$ when $3$ is added to both its numerator and its denominator. It becomes $\dfrac{1}{2}$ when $1$ is subtracted from both. Find the fraction.

Let the fraction be $\dfrac{x}{y}$.

• $\dfrac{x + 3}{y + 3} = \dfrac{2}{3}$ gives $3x + 9 = 2y + 6$, so $3x - 2y = -3$.
• $\dfrac{x - 1}{y - 1} = \dfrac{1}{2}$ gives $2x - 2 = y - 1$, so $2x - y = 1$.

From the second equation, $y = 2x - 1$. Substitute: $3x - 2(2x - 1) = -3$, so $-x + 2 = -3$ and $x = 5$. Then $y = 9$. The fraction is $\dfrac{5}{9}$. Check. $\dfrac{8}{12} = \dfrac{2}{3}$ and $\dfrac{4}{8} = \dfrac{1}{2}$.

Money, mixtures and the sides of a rectangle follow the same five steps.

Practise with new numbers

Try it in Python

The program finds the speeds of the boat and the stream from the two times, and checks them against the story.

Common mistakes

• Writing a two-digit number as $xy$. Its value is $10x + y$.
• Mixing up past and future ages. "Eight years ago" means subtract $8$ from every age. "In eight years" means add $8$.
• Using the wrong stream speed. Downstream the speed is $u + v$, and upstream it is $u - v$.
• Skipping the check against the words. Equations can be solved correctly and still give a digit of $12$ or a negative age.

Remember these

• Name the unknowns with their units, and write one equation for each fact.
• A two-digit number with tens digit $x$ and units digit $y$ is $10x + y$.
• Downstream speed is $u + v$. Upstream speed is $u - v$.
• Check the answer against the words of the problem, and answer with units.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two numbers add up to $40$, and their difference is $12$. Find them.

2. Sita is $3$ times as old as Gita. In $10$ years Sita will be $2$ times as old as Gita. Find their present ages.

3. A boat goes $9$ km/h in still water and the stream flows at $3$ km/h. How long does it take to go $24$ km upstream?

4. The tens digit of a number is $2$ more than its units digit, and the digits add to $10$. What is the number?

5. Why should you check a word problem against the words as well as against the equations?