Word problems
Set up and solve pairs of simultaneous equations from real-world problems. The key skill is translating the words into two equations, then choosing the most efficient method to solve them.
Worked examples
Sum and difference of two numbers
Straightforward
Problem
Two numbers add to 20 and their difference is 6. Find both numbers.
1
Define variables. Let the two numbers be and , where is the larger.
2
Add the equations to eliminate .
3
Substitute into the first equation.
4
Check and write a sentence answer.
Check: ✓ ✓
The two numbers are 13 and 7.
The two numbers are 13 and 7.
Answer
13 and 7
Ticket pricing problem
Moderate
Problem
Adult tickets cost $8 and student tickets cost $5. 120 tickets were sold for a total of $741. How many of each type were sold?
1
Define variables.
Let = number of adult tickets
Let = number of student tickets
Let = number of student tickets
2
Write two equations: one for the total number of tickets, one for the total revenue.
3
From the first equation, express in terms of and substitute.
4
Check: ✓
47 adult tickets and 73 student tickets were sold.
Find and check.
Check: ✓
47 adult tickets and 73 student tickets were sold.
Answer
47 adult tickets and 73 student tickets
Mobile plan — finding fixed fee and rate
Moderate
Problem
A mobile plan charges a fixed monthly fee plus a charge per GB of data. Sam used 3 GB in January and paid $42. He used 5 GB in February and paid $54. Find the fixed monthly fee and the cost per GB.
1
Define variables.
Let = fixed monthly fee
Let = cost per GB
Let = cost per GB
2
Write two equations from the two months.
3
Subtract the first equation from the second to eliminate .
4
Substitute into the first equation.
5
Check and state the answer.
Check February: ✓
Fixed fee: $24 per month. Data rate: $6 per GB.
Fixed fee: $24 per month. Data rate: $6 per GB.
Answer
Fixed fee $24, data rate $6 per GB
Practise
Q1·Straightforward
Two numbers add to 20 and their difference is 4. What is the larger number?
Explanation
Let be the larger number, the smaller.
Add the equations:
The larger number is 12.
Check: ✓, ✓
Add the equations:
The larger number is 12.
Check: ✓, ✓
Q2·Straightforward
Two numbers add to 35. One number is 7 more than the other. What is the larger number?
Explanation
Let the smaller number be . The larger is .
Larger number
Check: ✓, ✓
Larger number
Check: ✓, ✓
Q3·Straightforward
The perimeter of a rectangle is 36 cm. The length is 4 cm more than the width. What is the length in cm?
Explanation
Let = length, = width.
Half the perimeter:
Relationship:
Substitute:
Check: ✓
Half the perimeter:
Relationship:
Substitute:
Check: ✓
Q4·Straightforward
A class has 30 students. There are 6 more girls than boys. How many girls are in the class?
Explanation
Let = girls, = boys.
Substitute:
Check: ✓, ✓
Substitute:
Check: ✓, ✓
Q5·Moderate
Adult tickets to a concert cost $8 and student tickets cost $5. 120 tickets were sold for a total of $741. How many adult tickets were sold?
Explanation
Let = adult tickets, = student tickets.
From the first equation:
Substitute:
Check: ✓
From the first equation:
Substitute:
Check: ✓
Q6·Moderate
Jenny bought 7 chocolates for $27. Type A cost $3 each and Type B cost $5 each. How many Type B chocolates did she buy?
Explanation
Let = Type A, = Type B.
From the first:
Substitute:
Check: ✓
From the first:
Substitute:
Check: ✓
Q7·Moderate
A mobile plan charges a fixed monthly fee plus a cost per GB of data. In January Sam used 3 GB and paid $42. In February he used 5 GB and paid $54. What is the fixed monthly fee in dollars?
Explanation
Let = fixed fee, = cost per GB.
Subtract the first from the second:
Substitute into the first:
The fixed monthly fee is $24.
Subtract the first from the second:
Substitute into the first:
The fixed monthly fee is $24.
Q8·Moderate
A piggy bank contains 40 coins, all 10c and 50c pieces. The total value is $12.00. How many 50c coins are there?
Explanation
Let = number of 10c coins, = number of 50c coins.
From the first:
Substitute:
Check: c ✓
From the first:
Substitute:
Check: c ✓
Q9·Challenging
A boat travels 60 km downstream in 3 hours. The return trip upstream takes 5 hours. What is the speed of the current in km/h?
Explanation
Let = boat speed in still water, = current speed (both in km/h).
Downstream:
Upstream:
Simplify:
Add the equations:
Substitute into :
The current flows at 4 km/h.
Downstream:
Upstream:
Simplify:
Add the equations:
Substitute into :
The current flows at 4 km/h.
Q10·Challenging
A chemist needs 200 mL of a 45% acid solution. She has a 30% solution and a 50% solution. How many mL of the 50% solution should she use?
Explanation
Let = mL of 30% solution, = mL of 50% solution.
From the first equation:
Substitute:
Use 150 mL of the 50% solution (and 50 mL of the 30% solution).
From the first equation:
Substitute:
Use 150 mL of the 50% solution (and 50 mL of the 30% solution).
Q11·Challenging
The sum of the digits of a two-digit number is 9. When the digits are reversed, the new number is 27 more than the original. What is the original number?
Explanation
Let = tens digit, = units digit.
Digit sum:
Reversed is 27 more:
Solve the system:
Add the equations:
The original number is 36.
Check: ✓, reversed , ✓
Digit sum:
Reversed is 27 more:
Solve the system:
Add the equations:
The original number is 36.
Check: ✓, reversed , ✓
Q12·Challenging
Two friends, Alex and Ben, together have $200. If Alex gave Ben $30, they would have equal amounts. How much does Alex currently have?
Explanation
Let = Alex's amount, = Ben's amount.
Total:
After transfer: , so
Add the equations:
Alex currently has $130.
Check: ✓, after transfer: ✓
Total:
After transfer: , so
Add the equations:
Alex currently has $130.
Check: ✓, after transfer: ✓