A two-digit number's digits add to 10. Swapping its tens and ones digits gives a number 18 larger. What is the number?

46. Tens digit x, ones digit y: x + y = 10, (10y + x) − (10x + y) = 18, so y − x = 2. Then x = 4, y = 6: the number is 46.

A father is 5 times as old as his son. In 8 years he will be 3 times as old. How old is the son now?

8 years old. Father x, son y: x = 5y and x + 8 = 3(y + 8). Substituting, 5y + 8 = 3y + 3 × 8, so y = 8.

A boat goes 20 km downstream in 2 hours and the same way back upstream in 10 hours. Its speed in still water is □ km/h and the current's □ km/h. Fill in each blank

6, 4. Downstream u + v = 20 ÷ 2, upstream u − v = 20 ÷ 10. Add and halve: u = 6; subtract and halve: v = 4.

Simultaneous linear equations · Ages 13–14

Word problems with simultaneous equations

Choose, form, solve, then check it makes sense.

How to do it

Method

Choose, form, solve, then check it makes sense

x + y = 10, 2x + 4y = 32: x = 4, y = 6

Another way

Suppose them all alike

All chickens: 20 legs; (32 − 20) ÷ 2 = 6

In full

Four steps: choose two unknowns, form two equations, solve them, and check the answer makes sense (counts must be whole and positive). Chickens and rabbits: 10 animals, 32 legs. With x chickens and y rabbits: x + y = 10 and 2x + 4y = 32; halve the second and take away the first, y = 6, x = 4. Digit problems: a two-digit number with tens digit a and ones digit b is 10a + b, not ab. Or suppose them all alike: all chickens would be 20 legs, 12 short; each swap of a chicken for a rabbit adds 2 legs, so 12 ÷ 2 = 6 swaps. The same answer.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
A two-digit number's digits add to 10. Swapping its tens and ones digits gives a number 18 larger. What is the number?
Show the answer and working
Answer: 46
Tens digit x, ones digit y: x + y = 10, (10y + x) − (10x + y) = 18, so y − x = 2. Then x = 4, y = 6: the number is 46.
Example 2
A father is 5 times as old as his son. In 8 years he will be 3 times as old. How old is the son now?
Show the answer and working
Answer: 8 years old
Father x, son y: x = 5y and x + 8 = 3(y + 8). Substituting, 5y + 8 = 3y + 3 × 8, so y = 8.
Example 3
A boat goes 20 km downstream in 2 hours and the same way back upstream in 10 hours. Its speed in still water is km/h and the current's km/h.
Fill in each blank
Show the answer and working
Answer: 6, 4
Downstream u + v = 20 ÷ 2, upstream u − v = 20 ÷ 10. Add and halve: u = 6; subtract and halve: v = 4.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Chickens and rabbits: 33 heads and 92 legs. How many rabbits are there?
Answer:
2.$5 and $2 coins, 18 in all, worth $75. There are $5 coins and $2 coins.
Fill in each blank
Answer:
3.Two numbers add to 128 and differ by 26. The larger is and the smaller .
Fill in each blank
Answer:
4.A boat goes 64 km downstream in 4 hours and the same way back upstream in 8 hours. Its speed in still water is km/h and the current's km/h.
Fill in each blank
Answer:
5.A father is 5 times as old as his son. In 3 years he will be 4 times as old. How old is the son now?
Answer:

All the answers

1. 13 rabbits
2. 13, 5
3. 77, 51
4. 12, 4
5. 9 years old

Free worksheet: Word problems with simultaneous equationsWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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