Printable · GCSE Foundation · ages 14-16
Expressions, equations, formulae, identities and inequalities worksheet — GCSE Foundation
Fifteen questions on "expressions, equations, formulae, identities and inequalities" — DfE statement A3. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Answer key: Expressions, equations, formulae, identities and inequalities worksheet — GCSE Foundation
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- (b) 3w − 5 ≥ 16 — "Three times w, minus 5" translates to 3w − 5, and "is at least 16" means it must be 16 or more, giving 3w − 5 ≥ 16. A candidate who reads "at least" as a strict inequality writes 3w − 5 > 16. A candidate who misreads the wording and applies the subtraction before the multiplication writes 3(w − 5) ≥ 16. A candidate who reverses the direction of the inequality writes 3w − 5 ≤ 16.
- (d) 2n + 7 > 15 — An inequality compares two expressions using a symbol other than an equals sign, such as < or >. "2n + 7" is just an expression — it has no equals or inequality sign at all, so a candidate mistaking any bare expression for an inequality would choose this. "2n + 7 = 15" is an equation: it uses an equals sign and is only true for one particular value of n, so a candidate confusing an equation with an inequality would choose this. "P = 2n + 7" is a formula, since it relates two different letters, P and n, with an equals sign, so a candidate confusing a formula with an inequality would choose this. "2n + 7 > 15" is an inequality: it compares an expression with a number using a > sign.
- (b) An identity, since both sides are equal for every x. — Expanding the left-hand side: 5(2x−3)=10x−15, which is exactly the same as the right-hand side for every value of x, so it is an identity, not an equation that is only true for one particular x. A candidate who treats every equals-sign statement as an equation, without checking whether it holds for all values of x, would choose the equation option. A candidate who mistakes it for a formula is assuming it relates two different letters or quantities, but only x appears — there is no second variable such as area or cost — so it is not a formula. A candidate who mistakes it for an inequality is assuming the two sides are only equal for particular values of x, but expanding shows they are identical for every value, not just some.
- (a) The equation has one solution; the inequality has many. — Method: solve each statement. From 2x + 3 = 11, 2x = 8, so x = 4 — a single value. From 2x + 3 > 11, 2x > 8, so x > 4 — every number greater than 4 makes the inequality true, so there are many solutions. So the equation has one solution and the inequality has many. Distractor origins: swapping the two round gives the range to the equation and the single value to the inequality; saying both have exactly one solution treats the > sign as if it were an = sign; saying both have many solutions treats the equation as if it were an inequality.
- (d) An equation, true for one value of x — 2x + 5 = 17 is only true when x = 6, so it is an equation. A candidate who ignores the equals sign and focuses only on the x terms and numbers present picks the expression option. A candidate who sees an equals sign and assumes it must relate two different quantities, as a formula does, picks that option, missing that there is only one quantity, x, involved here. A candidate who wrongly checks whether both sides look similar in structure rather than testing specific values picks the identity option, missing that the two sides are not equal for every value of x.
- (d) an equation — Method: recall what each piece of algebraic vocabulary means and match it to the description given. Working: a statement with an equals sign, one unknown letter, and only particular value(s) making it true is, by definition, an equation. Answer: an equation. A formula also has an equals sign, but it links two or more different letters so that one quantity can be calculated from the others — here there is only one letter. An identity is a statement with an equals sign that is true for every value of the letter, not just particular ones. An expression has no equals sign at all, so it cannot be 'true' or 'false'.
- (b) £115, and C = 40 + 25h is a formula — Substitute h = 3 into the rule, multiplying before adding. The hours cost 25 × 3 = 75, and adding the call-out fee gives 40 + 75 = 115, so the charge is £115. The rule itself links two different quantities, C and h, and lets one be worked out from the other, so it is a formula; an expression would have no equals sign in it. Adding the fee before multiplying gives 65 × 3 = 195, which charges the call-out fee three times over, and stopping at 40 + 25 leaves £65, the charge for a single hour.
- (b) P = 4s — A formula is a rule connecting two or more different quantities, so that one of them can be worked out once the others are known. P = 4s does exactly that: it gives P from s. The line with no equals sign is an expression; the line with an equals sign and a single unknown letter is an equation, because it is solved to find that letter; the line joined by a comparison sign is an inequality.
- (b) 12 — Method: substitute the total cost into the formula, then subtract the fixed cost and divide by the cost per person. Working: 92 = 6p + 20, so 6p = 92 − 20 = 72, p = 72 ÷ 6 = 12. Answer: 12 people. 15.33 comes from dividing the whole £92 by £6 without first subtracting the £20 fixed cost: 92 ÷ 6 ≈ 15.33. 4.3 comes from swapping the two amounts round, subtracting £6 and dividing by £20: (92 − 6) ÷ 20 = 4.3. 18.67 comes from adding the fixed cost instead of subtracting it: (92 + 20) ÷ 6 ≈ 18.67.
- (a) 5 — A factor of the whole expression must divide every term exactly. 15x + 20 = 5(3x + 4), so 5 is a factor. Distractor origins: 3 divides 15x exactly but does not divide 20 exactly; 4 divides 20 exactly but does not divide 15x exactly; 10 also divides 20 exactly but does not divide 15x exactly, so it is a factor of only one term, not of the whole expression.
- (c) 5x² — A term is one part of an expression, separated from the others by a + or − sign. The expression 5x² − 3x + 7 has three terms: 5x², −3x and 7. Distractor origins: 5x² − 3x combines two terms together, not a single term; −3x + 7 also combines two terms together; 5x² − 3x + 7 gives the whole expression, not a single term.
- (b) £47, and T=15+8h is a formula (relates T and h). — The charge is £15 fixed plus £8 for each hour: T=15+8h, so for h=4, T=15+8×4=15+32=47. Because T=15+8h relates two different quantities, the total charge T and the number of hours h, it is a formula, not an equation — an equation is solved for one particular value of an unknown, but this relationship holds for every value of h a job might last. A candidate who forgets to include the fixed £15 fee would compute only 8×4=32. A candidate who adds the three numbers in the formula together instead of multiplying the hourly rate by the number of hours would compute 15+8+4=27. A candidate who correctly finds the charge but mistakes the formula for an equation, treating it as something to be solved for one specific value of h rather than a general relationship between T and h, would pick the correct charge with the wrong classification.
- (d) 5x² — 5x² is a single term: a product of a number and a power of a letter, with no + or − sign attached to anything else. A candidate who picks 5x² + 3x is choosing the expression that contains that term, rather than the term on its own. A candidate who picks 5x² = 20 is drawn in by the equals sign and misses that this makes it an equation. A candidate who picks A = 5x² thinks any statement with a letter and a number combined must be a formula.
- (a) An identity, because it is true for every value of x — Expanding the brackets multiplies both terms inside by 4, giving 4x + 12, which is exactly the right-hand side. The two sides are therefore equal whatever x is, and a statement true for every value of the letter is an identity. An equation is true only for particular values, and trying to solve this one leads to 0 = 0, which places no restriction on x at all. Being able to expand brackets is not what makes a statement an identity, since any equation with brackets can be expanded. Nor is it a formula: a formula links two different quantities, and only one letter appears here.
- (d) 3 — The expression has three terms separated by + and − signs: 4x², −7x and 3. A candidate who thinks a number on its own does not count as a term says there are 2 terms, missing the 3. A candidate who miscounts the minus sign as creating an extra term says there are 4. A candidate who splits each term into its coefficient and letter part separately (4, x², 7, x, 3) says there are 5.
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