Formulae and identities – Year 10 GCSE maths worksheet with answers to download as PDF

Calculator allowed. Use the given values to work out each answer.

1. \(v=u+at\) with \(u=5\), \(a=2.5\), \(t=6\). Find \(v\).
2. \(s=ut+\tfrac12at^2\) with \(u=3\), \(a=4\), \(t=5\). Find \(s\).
3. \(A=\tfrac12(a+b)h\) with \(a=7\), \(b=11\), \(h=6\). Find \(A\).
4. \(v^2=u^2+2as\) with \(u=4\), \(a=3\), \(s=10\). Find \(v\) (positive value).

Make the letter in brackets the subject of each formula.

1. \(y=mx+c\) (make \(x\) the subject)
2. \(P=2l+2w\) (make \(l\) the subject)
3. \(V=lwh\) (make \(h\) the subject)
4. \(C=2\pi r\) (make \(r\) the subject)

Make the letter in brackets the subject. Show each step.

1. \(A=\tfrac12(a+b)h\) (make \(a\) the subject)
2. \(v^2=u^2+2as\) (make \(a\) the subject)
3. \(s=\tfrac12(u+v)t\) (make \(v\) the subject)
4. \(d=\sqrt{2h}\) (make \(h\) the subject)

Decide whether each is an expression, an equation, a formula or an identity. Give a reason.

1. \(3x+5=20\)
2. \(A=\pi r^2\)
3. \((x+2)(x-2)\equiv x^2-4\)
4. \(4a-7b+2\)

Show that each identity is true by expanding and simplifying one side.

1. \(2(x+4)+3(x-1)\equiv5x+5\)
2. \((x+3)^2-(x-3)^2\equiv12x\)
3. \((x+5)(x-2)\equiv x^2+3x-10\)
4. \((2x-1)^2\equiv4x^2-4x+1\)

Make \(x\) the subject of each formula. Give both positive and negative roots where they apply.

1. \(y=3x^2-5\)
2. \(y=\dfrac{x^2}{4}+1\)
3. \(A=\pi x^2\) (positive \(x\) only)
4. \(y=(x+2)^2\)

In the UK, weather forecasts use degrees Celsius (°C). Some visitors use degrees Fahrenheit (°F). The formula is \(F=1.8C+32\).

1. Convert each temperature to °F.
2. Make \(C\) the subject of the formula.
3. A US visitor’s oven is set to 350 °F. Find the temperature in °C to the nearest degree.
4. At what temperature are the Celsius and Fahrenheit values equal?

An energy company charges £\(C\) per month by the formula \(C=0.28u+0.45d\) where \(u\) is the number of units of electricity used and \(d\) is the number of days. The standing charge is built into the second term.

1. Work out the bill for 160 units in a 30-day month.
2. Make \(u\) the subject of the formula.
3. A customer in a 31-day month has a bill of £70. Find the units used, to the nearest unit.
4. Another firm charges £\(C=0.25u+0.62d\). For 160 units over 30 days, which firm is cheaper, and by how much?

The cooking time \(T\) minutes for a joint of beef of mass \(w\) kg is given by \(T=40w+20\).

1. Work out the cooking time for a 2.5 kg joint, in hours and minutes.
2. Make \(w\) the subject of the formula.
3. Dinner is planned for 6:30 pm. The joint weighs 3.2 kg. At what time should it go in the oven?
4. What is the largest joint that can be cooked in \(2\tfrac12\) hours?

A stone is dropped from a bridge. The distance fallen, \(s\) metres, after \(t\) seconds is \(s=\tfrac12gt^2\) where \(g=9.8\) m/s².

1. How far does the stone fall in 3 seconds?
2. Make \(t\) the subject of the formula.
3. The bridge is 78.4 m above the river. How long does the stone take to reach the water?
4. The stone takes 2 seconds to reach the water from a different bridge. How high is that bridge?

Body mass index is \(B=\dfrac{m}{h^2}\) where \(m\) is mass in kg and \(h\) is height in metres. A school nurse uses it in a health talk.

1. Work out \(B\) for \(m=70\) and \(h=1.75\), to 1 decimal place.
2. Make \(m\) the subject.
3. Find the mass for \(B=24\) and \(h=1.6\).
4. Make \(h\) the subject of the formula and find \(h\) when \(m=63\) and \(B=25\).

The value of a savings account after \(n\) years is \(A=P\left(1+\dfrac{r}{100}\right)^n\), where \(P\) is the amount invested and \(r\) is the yearly percentage rate. Calculator allowed.

1. Find \(A\) when \(P=500\), \(r=4\) and \(n=3\), to the nearest penny.
2. Find \(A\) when \(P=1200\), \(r=2.5\) and \(n=5\), to the nearest penny.
3. Simple interest is \(I=\dfrac{PRT}{100}\). Make \(T\) the subject.
4. Find the time for £800 to earn £120 simple interest at 3% per year.

The cross-section of a swimming pool is a trapezium with parallel sides \(a\) (the shallow end) and \(b\) (the deep end) and a length \(h\) (the length of the pool). The area of the cross-section is \(A=\tfrac12(a+b)h\).

1. Find the area when \(a=1\) m, \(b=2.5\) m and \(h=20\) m.
2. Make \(b\) the subject of the formula.
3. The area is 48 m² with \(a=1.2\) m and \(h=20\) m. Find \(b\).
4. The pool’s width is 8 m. Find the volume of water in the pool in the situation from part (a).

A cylindrical paint tin has volume \(V=\pi r^2h\).

1. Find \(V\) for \(r=7\) cm and \(h=10\) cm, to the nearest cm³.
2. Make \(r\) the subject of the formula.
3. A tin must hold exactly 750 cm³ and has height 12 cm. Find \(r\) to 1 decimal place.
4. Make \(h\) the subject and find \(h\) for a tin with radius 6 cm and volume 1000 cm³, to 1 d.p.

Higher tier. When the new subject appears in more than one place, collect all the terms containing it on one side, then factorise.

1. Make \(x\) the subject of \(ax+b=cx+d\).
2. Make \(x\) the subject of \(y=\dfrac{2x+1}{x-3}\).
3. Make \(a\) the subject of \(S=\dfrac{a}{1-r}\) (\(r\) is not 1).
4. Make \(p\) the subject of \(p(q+1)=3(p-2)\).

Use algebra to prove each statement. Remember that you must not use numbers as proof, only as a check.

1. Prove that \((n+1)^2-(n-1)^2\equiv4n\).
2. Prove that the sum of the squares of two consecutive whole numbers is always odd.
3. Prove that \((2n+1)^2-1\) is always a multiple of 4. (Hint: expand and factorise.)
4. Show that \((2n+1)^2-1\) is actually a multiple of 8.

Higher tier. The stopping distance \(d\) metres of a car travelling at \(v\) miles per hour is modelled by \(d=\dfrac{v^2}{60}+0.3v\) (this is a simplified model).

1. Find the stopping distance at 30 mph.
2. Find the stopping distance at 70 mph, to the nearest metre.
3. Show that \(d=60\) gives \(v^2+18v-3600=0\).
4. Solve this equation to find the speed that gives a stopping distance of 60 m, to 1 decimal place.

An identity is true for all values of \(x\), so the coefficients on each side must match.

1. Find \(a\) and \(b\) if \((x+a)^2+b\equiv x^2+6x+13\).
2. Find \(p\) and \(q\) if \(p(x+3)+q\equiv5x+19\).
3. Find \(k\) if \((x+k)(x-4)\equiv x^2+2x-24\).
4. Show that there are no values of \(a\) and \(b\) such that \(2(x+a)\equiv2x+b+x\) for all \(x\).

The formula for the length \(L\) of an arc of a circle of radius \(r\) and angle \(\theta\) degrees is \(L=\dfrac{\theta}{360}\times2\pi r\).

1. Make \(\theta\) the subject.
2. Make \(r\) the subject.
3. Find \(\theta\) when \(L=15.7\) and \(r=10\), to the nearest degree.
4. Find \(r\) when \(L=22\) and \(\theta=90\), to 1 decimal place.

The \(n\)th triangular number is \(T_n=\dfrac{n(n+1)}{2}\). The first few are 1, 3, 6, 10, 15 (dots arranged in a triangle).

1. Find \(T_6\) and \(T_{10}\).
2. Show that \(T_n+T_{n+1}=(n+1)^2\).
3. Use part (b) to explain why the sum of two consecutive triangular numbers is always a square number.
4. Find \(n\) if \(T_n=120\).