The ENGAA is a heavily time-pressured exam, particularly Section 1. I would like to help students combat this and have the following advice. Some of the ideas mentioned below can be practised using material from my eBook.

Get into the habit of working without a calculator unless it is absolutely necessary. It is surprising what can be achieved without one and attempting to do this will sharpen your mental arithmetic.

Once you have identified ways in which you can save time, put this into practice by attempting many BMAT questions. Ignore any subjects that are unnecessary for your exam such as Biology or Chemistry. The style of ENGAA/NSAA questions is similar to that of BMAT and there are a great number of BMAT past papers available. When practising try to do the entire question where possible by inspection or by only writing down one or two interim steps.

This seems counter-intuitive but in my opinion most students will run out of time and you still stand a good chance of getting an offer even if you don't reach the last 10 questions. To convince yourself that you have to get to the end of the Section no matter what will could lead to excessive clock-watching and a flustered approach. Do your best to complete the questions efficiently and if you get to the end then that's a bonus.

If you are stuck on a question it is best to move on and harvest easier marks elsewhere in the paper. You can always come back to questions you had difficulty with the first time around. Also, be prepared to spend less time on the earlier questions because they are easier.

This can be a very powerful strategy. Sometimes a small amount of working can reveal some or all of the incorrect multiple choice answers. On occasion the answers may be very far apart numerically allowing for crude approximations in the arithmetic.

Solving quadratic equations is quite common in the ENGAA. Firstly, be on the lookout for the opportunity to divide a quadratic by a number to make the coefficients smaller as follows:

$5x^2+10x-120=5(x^2+2x-24)=5(x+6)(x-4)$

If you can't see a factorisation immediately then use the formula if necessary. Looking at the answers can also give you clues as to how the equation could be factorised.

$3$,$4$,$5$ Right angled triangle.

Half an equilateral triangle (sides $1$, $\sqrt3$, $2$. Angles $30$, $60$, $90$)

Isosceles right angled triangle (sides $1$, $1$, $\sqrt2$. Angles $45$, $45$, $90$)

These triangles are very common in non-calculator papers at this level. It pays to know them well and to be on the lookout for scalar multiples of them.

It is very common for the numerator of one fraction to cancel with the denominator of another as follows:

$\frac{8}{9}\times\frac{5}{24}=\frac{1}{9}\times\frac{5}{3}=\frac{5}{27}$

This idea can be taken a lot further in exams such as the ENGAA. I very rarely perform any multiplications until the last step and always look for cancellations where possible, especially when I am dealing with factor-rich numbers such as 24, 36, 72 etc.

Be prepared to swap between the two different forms. Here are two examples of cases where it is significantly easier to use one rather than the other.

$a^2-b^2=(a+b)(a-b)$

$3.75^2-0.25^2=(3.75+0.25)(3.75-0.25)=4\times3.5=14$

$9.1\times8.9=(9+0.1)(9-0.1)=81-0.01=80.99$

If you have a fraction which involves a quadratic equation on the numerator and denominator, factorise the simpler one first as this will give you a clue as to how the more complicated one might factorise.

This is more personal preference, but you may find it easier to say divide by 3 four times than divide by 81.

Rearrange so that the quantity you have to calculate is the subject before substituting numbers in. This will help you to spot cancellations more readily.

If you have a fractional power, then it is best to perform the operation which reduces the size of the number first followed by the operation which increases the size of it:

$(\frac{27}{8})^{\frac{2}{3}}=((\frac{27}{8})^{\frac{1}{3}})^2=(\frac{3}{2})^2=\frac{9}{4}$

is better than:

$(\frac{27}{8})^{\frac{2}{3}}=((\frac{27}{8})^{2})^{\frac{1}{3}}=(\frac{729}{64})^{\frac{1}{3}}=\frac{9}{4}$

The idea of choosing the order of operations to work with smaller numbers rather than larger ones is not limited to fractional powers; it is a good general principle to work from when trying to save time.

Work in the units of the answers - for example if the answers are in hours then you can avoid an unnecessary multiplication and division by $3600$ by keeping to hours as opposed to converting to seconds.

You should be able to convert between $0.25=\frac{1}{4}$, $0.2=\frac{1}{5}$, $0.125=\frac{1}{8}$ and multiples of them e.g. $1.25=\frac{5}{4}$.

You may be able to easily see that your answer starts with a certain digit but counting the zeros at the end of the number might waste some time. If all the answers start with different digits then you can avoid spending time on working out the order of magnitude.

Be aware that it may be quicker to work out $1$ minus the probability of your event not happening than to work out the probability of your event happening directly.

If asked to evaluate something, see if it's possible to re-write/simplify it to your advantage before you begin to work it out.

It is common for low powers of numbers to crop up, both in the examiner's questions and the students' working. Examples would be powers of $2$ up to $64$, powers of $3$ up to $243$, powers of $5$ up to $625$, cube numbers up to $343$ and square numbers up to $289$.

For questions where you have to rearrange a formula to make a certain variable the new subject, you may wish to try substituting in a value or some values for one or both of the variables to reveal the answer (Thank you to Martin C for this idea).

Complete the paper with a pencil so that you don't have to swap writing implements between working out and shading in the relevant circle on the answer sheet (Thank you to Martin C for this idea).

Thank you for reading - now I recommend you put the above into practice by practising with BMAT papers (Section 2). Good luck!

**1. Change the way you solve problems between now and the ENGAA.**Get into the habit of working without a calculator unless it is absolutely necessary. It is surprising what can be achieved without one and attempting to do this will sharpen your mental arithmetic.

**2. Use past BMAT problems (Section 2) to practise and try to do more steps by inspection.**Once you have identified ways in which you can save time, put this into practice by attempting many BMAT questions. Ignore any subjects that are unnecessary for your exam such as Biology or Chemistry. The style of ENGAA/NSAA questions is similar to that of BMAT and there are a great number of BMAT past papers available. When practising try to do the entire question where possible by inspection or by only writing down one or two interim steps.

**3. Accept that you might not get to the end of Section 1.**This seems counter-intuitive but in my opinion most students will run out of time and you still stand a good chance of getting an offer even if you don't reach the last 10 questions. To convince yourself that you have to get to the end of the Section no matter what will could lead to excessive clock-watching and a flustered approach. Do your best to complete the questions efficiently and if you get to the end then that's a bonus.

**4. Be prepared to leave questions that are causing you difficulty.**If you are stuck on a question it is best to move on and harvest easier marks elsewhere in the paper. You can always come back to questions you had difficulty with the first time around. Also, be prepared to spend less time on the earlier questions because they are easier.

**5. Keep an eye on the answers.**This can be a very powerful strategy. Sometimes a small amount of working can reveal some or all of the incorrect multiple choice answers. On occasion the answers may be very far apart numerically allowing for crude approximations in the arithmetic.

**6. Learn to factorise quadratics quickly.**Solving quadratic equations is quite common in the ENGAA. Firstly, be on the lookout for the opportunity to divide a quadratic by a number to make the coefficients smaller as follows:

$5x^2+10x-120=5(x^2+2x-24)=5(x+6)(x-4)$

If you can't see a factorisation immediately then use the formula if necessary. Looking at the answers can also give you clues as to how the equation could be factorised.

**7. Memorise three familiar triangles.**$3$,$4$,$5$ Right angled triangle.

Half an equilateral triangle (sides $1$, $\sqrt3$, $2$. Angles $30$, $60$, $90$)

Isosceles right angled triangle (sides $1$, $1$, $\sqrt2$. Angles $45$, $45$, $90$)

These triangles are very common in non-calculator papers at this level. It pays to know them well and to be on the lookout for scalar multiples of them.

**8. Look for cancellations and don't perform any unnecessary calculations.**It is very common for the numerator of one fraction to cancel with the denominator of another as follows:

$\frac{8}{9}\times\frac{5}{24}=\frac{1}{9}\times\frac{5}{3}=\frac{5}{27}$

This idea can be taken a lot further in exams such as the ENGAA. I very rarely perform any multiplications until the last step and always look for cancellations where possible, especially when I am dealing with factor-rich numbers such as 24, 36, 72 etc.

**9. Difference of two squares.**Be prepared to swap between the two different forms. Here are two examples of cases where it is significantly easier to use one rather than the other.

$a^2-b^2=(a+b)(a-b)$

$3.75^2-0.25^2=(3.75+0.25)(3.75-0.25)=4\times3.5=14$

$9.1\times8.9=(9+0.1)(9-0.1)=81-0.01=80.99$

**10. Factorise strategically**If you have a fraction which involves a quadratic equation on the numerator and denominator, factorise the simpler one first as this will give you a clue as to how the more complicated one might factorise.

**11. Embrace the percentage multiplier method and try this first.****12. Division by a power.**This is more personal preference, but you may find it easier to say divide by 3 four times than divide by 81.

**13. Rearrange first.**Rearrange so that the quantity you have to calculate is the subject before substituting numbers in. This will help you to spot cancellations more readily.

**14. Order of fractional power operations.**If you have a fractional power, then it is best to perform the operation which reduces the size of the number first followed by the operation which increases the size of it:

$(\frac{27}{8})^{\frac{2}{3}}=((\frac{27}{8})^{\frac{1}{3}})^2=(\frac{3}{2})^2=\frac{9}{4}$

is better than:

$(\frac{27}{8})^{\frac{2}{3}}=((\frac{27}{8})^{2})^{\frac{1}{3}}=(\frac{729}{64})^{\frac{1}{3}}=\frac{9}{4}$

The idea of choosing the order of operations to work with smaller numbers rather than larger ones is not limited to fractional powers; it is a good general principle to work from when trying to save time.

**15. Note the units in the answers.**Work in the units of the answers - for example if the answers are in hours then you can avoid an unnecessary multiplication and division by $3600$ by keeping to hours as opposed to converting to seconds.

**16. Know simple fraction to decimal conversions.**You should be able to convert between $0.25=\frac{1}{4}$, $0.2=\frac{1}{5}$, $0.125=\frac{1}{8}$ and multiples of them e.g. $1.25=\frac{5}{4}$.

**17. Avoid counting unnecessary zeros.**You may be able to easily see that your answer starts with a certain digit but counting the zeros at the end of the number might waste some time. If all the answers start with different digits then you can avoid spending time on working out the order of magnitude.

**18. Take the quickest route with probabilities.**Be aware that it may be quicker to work out $1$ minus the probability of your event not happening than to work out the probability of your event happening directly.

**19. Look for simplifications from the outset.**If asked to evaluate something, see if it's possible to re-write/simplify it to your advantage before you begin to work it out.

**20. Know low powers of small numbers.**It is common for low powers of numbers to crop up, both in the examiner's questions and the students' working. Examples would be powers of $2$ up to $64$, powers of $3$ up to $243$, powers of $5$ up to $625$, cube numbers up to $343$ and square numbers up to $289$.

For questions where you have to rearrange a formula to make a certain variable the new subject, you may wish to try substituting in a value or some values for one or both of the variables to reveal the answer (Thank you to Martin C for this idea).

Complete the paper with a pencil so that you don't have to swap writing implements between working out and shading in the relevant circle on the answer sheet (Thank you to Martin C for this idea).

**If you have any time saving strategies that you would like to share with other candidates, please contact me and I will post them here and credit your contribution.**Thank you for reading - now I recommend you put the above into practice by practising with BMAT papers (Section 2). Good luck!