SPI Mathematics · Linear Equations

Word Problems on Linear Equations

A word problem is solved by naming the unknown, writing each sentence as an equation, and answering the quantity the question actually asks for.

All Linear Equations notes

Why this matters

Seven questions in the bank, 4 EASY and 3 MODERATE, and four of them are from the 2022 paper, more than any other part of this chapter. The algebra is short. The work is choosing the unknown and writing each sentence as an equation.

Concept 1 of 3: One unknown from a sentence

Call the smallest or the most-mentioned quantity x, and write every other quantity in terms of x. Then the one fact about the total gives one equation.

Definition

'3 more than x' is $x + 3$ , '2 less than x' is $x - 2$ , 'twice x' is $2x$ , 'half of x' is $\frac{x}{2}$ .
A fixed part plus a rate gives a linear rule: a fixed charge of ₹50 plus ₹8 per km is $y = 50 + 8x$ .
Answer the quantity asked for, which may not be x itself.

Fixed part plus a rate

y = a + bx \quad (a = \text{fixed part},\ b = \text{rate per unit}) \qquad x + (x + 1) + (x + 2) = 3x + 3

Worked example

A rope 46 m long is cut into three pieces. The second piece is 4 m longer than the first, and the third is 2 m longer than the second. Find the longest piece.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q58Easy

Example 1 · Linear Equations · Word Problems on Linear Equations

A farmer divided his 32 acres of land amongst his two son's and a daughter in such a manner that the elder son got 3 acres land more than the younger son. The daughter was given 2 acres of land more than the elder son. So the daughter got____acres.

Stopping at x

If x is the smallest piece and the question asks for the longest, work out the longest piece's expression. The value of x itself is often printed as a wrong option.

Reversing 'more than'

'A has 5 more than B' is

A = B + 5

, not

B = A + 5

. Decide who has more before you write the equation.

Drill 1 more on one unknown from a sentence

Concept 2 of 3: Chains of comparisons: ages and heights

When several people are compared one after another, pick the person who links the most statements and write everyone else in terms of that person. The whole chain then becomes one equation.

Definition

'A is half of B' means $B = 2A$ . 'A is one and a half times B' means $A = 1.5B$ .
'A is 12 years older than B' means $A = B + 12$ .
'Taller than M by the same amount as shorter than T' puts the height halfway between: $s - m = t - s$ , so $m + t = 2s$ .
Keep one unit: change metres to centimetres before adding a length in centimetres.
Check the answer against every statement, not just the last one.

Comparison sentences as equations

A = \tfrac{1}{2}B \;\Rightarrow\; B = 2A \qquad A = B + k \ (A \text{ is } k \text{ older}) \qquad s - m = t - s \;\Rightarrow\; m + t = 2s \ (s \text{ halfway})

Worked example

Ravi is twice as old as Meena. Meena is 4 years older than Joy. Ravi is 26 years older than Joy. How old is Joy?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q18Moderate

Example 2 · Linear Equations · Word Problems on Linear Equations

A group of five persons, namely Yaser, Seema, Dattu, Anand and Farhan are traveling together in a bus. Yaser's age is half that of Anand. Dattu's is one and half times that of Farhan. Yaser's age is one and half times that of Seema and Seema is 12 years elder to Dattu. The difference between Anand's age and that of Seema is 60 years. Then Farhan is ____ years old.

Mixing metres and centimetres

2.5 m is 250 cm. Adding 30 cm to a height still written as 2.5 gives nonsense, so change units first.

Reading 'half that of' backwards

'P's age is half that of Q' means

P = \frac{Q}{2}

, so Q is the older. Writing

Q = \frac{P}{2}

swaps the two people.

Drill 1 more on chains of comparisons: ages and heights

Concept 3 of 3: Two unknowns, two equations

When a question talks about two different counts, give each its own letter. Each fact becomes one equation, and the pair is solved by elimination.

Definition

Coins: for ₹1 and ₹2 coins, the number of coins is $x + y$ and their value is $x + 2y$ .
Heads and legs: for cows and hens, heads are $c + h$ and legs $4c + 2h$ .
Fractions: call the fraction $\frac{x}{y}$ . 'Add 1 to both' gives $\frac{x + 1}{y + 1}$ . Cross-multiply to get a linear equation.
Sometimes one unknown cancels and the question only needs the other.

Worked example

A purse holds ₹5 and ₹10 coins, 30 coins in all, worth ₹220. How many ₹10 coins are there?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q64Easy

Example 3 · Linear Equations · Word Problems on Linear Equations

In a group of cows and hens, the number of legs are 14 more than twice the number of heads. Then there are ____ cows.

Using one equation for two facts

The number of coins and their value are two different facts. One equation adds the coins; the other adds what each coin is worth.

Adding to the top only

'1 is added to both the numerator and the denominator' changes

\frac{x}{y}

\frac{x + 1}{y + 1}

. Adding to the numerator alone answers a different question.

Drill 2 more on two unknowns, two equations

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

One unknown from a sentence
Fixed part plus a rate
$y = a + bx \quad (a = \text{fixed part},\ b = \text{rate per unit}) \qquad x + (x + 1) + (x + 2) = 3x + 3$
Chains of comparisons: ages and heights
Comparison sentences as equations
$A = \tfrac{1}{2}B \;\Rightarrow\; B = 2A \qquad A = B + k \ (A \text{ is } k \text{ older}) \qquad s - m = t - s \;\Rightarrow\; m + t = 2s \ (s \text{ halfway})$

Watch out for (6)

Stopping at x
→ One unknown from a sentence
Reversing 'more than'
→ One unknown from a sentence
Mixing metres and centimetres
→ Chains of comparisons: ages and heights
Reading 'half that of' backwards
→ Chains of comparisons: ages and heights
Using one equation for two facts
→ Two unknowns, two equations
Adding to the top only
→ Two unknowns, two equations

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